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t1 angles and trigonometric functions

M
math260

There are two main systems for measuring angles: the degree system and radian system. The degree system divides a full rotation into 360 equal degrees, while the radian system defines an angle as the arc length cut out on a unit circle. There are also two important types of right triangles used in trigonometry: the 45-45-90 triangle where the two legs have length a and the hypotenuse has length a√2, and the 30-60-90 triangle where one leg has length a, the other has length a/2, and the hypotenuse has length 2a.

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Angular Measurements
Angular MeasurementsThere are two systems of angular measurements
Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system
Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o.
Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'.
Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1".
Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1". The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering.
Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1". The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering. In mathematics, the radian system is used more often because it's relationship with the geometry of circles.
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.r = 1
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Its given by the equation x2 + y2 = 1.r = 1
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems180o = π rad
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems180o = π rad 1o = rad π180
Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems180o = π rad 1o = rad = 1 rad  57oπ180oπ180
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position.
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is –
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), (x , y)(1,0)
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)(1,0)yx
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)y=sin()(1,0)x=cos()yx
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)y=sin()(1,0)x=cos()yxNote: tan() = slope of the dial
Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)tan()y=sin()(1,0)x=cos()yxNote: tan() = slope of the dial
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. π/4
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. π/4π/4π/4
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, π/4π/4π/4
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2π/4π/4π/4
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. π/6
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. π/6π/6c c = 2aπ/3π/3
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. π/6π/6c c = 2aπ/3π/3
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2π/6π/6c c = 2aπ/3π/3
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 a2 + b2 = 4a2 b2 = 3a2π/6π/6c c = 2aπ/3π/3
Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 a2 + b2 = 4a2 b2 = 3a2 b = 3a2 or b = a3π/6π/6c c = 2aπ/3π/3
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/412/2π/42/2
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric Values
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric ValuesThe trig-values of angles depend on the positions of the angle.
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric ValuesThe trig-values of angles depend on the positions of the angle. Following are the important positions of the dials for which we may find the exact trig-values.
Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric ValuesThe trig-values of angles depend on the positions of the angle. Following are the important positions of the dials for which we may find the exact trig-values. Note that  and  + 2nπhave the same position where n is an integer.
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.0, ±2π, ±4π..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.Kπ2
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..Angles with measurements of rad are diagonals.Kπ Frank Ma20064
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma20064
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ4
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200645π/4, -3π/4..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..Angles with measurements of (reduced) orrad.KπKπ63
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..Angles with measurements of (reduced) orrad.π/6, -11π/6..KπKπ63
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..Angles with measurements of (reduced) orrad.π/6, -11π/6..KπKπ63
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ63
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ7π/6,..634π/3,..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ7π/6,..6311π/6,..4π/3,..5π/3,..
Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ7π/6,..6311π/6,..4π/3,..5π/3,..
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3π
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3π-3π(-1, 0)
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/45π/4
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,5π/4
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,15π/4
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,we get the coordinate = (-2/2, -2/2).15π/4(-2/2, -2/2)
Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,we get the coordinate = (-2/2, -2/2).sin(5π/4) = -2/2 cos(5π/4) = -2/2 15π/4(-2/2, -2/2) tan(5π/4) = 1
Important Trigonometric Valuesc.  = -11π/6
Important Trigonometric Valuesc.  = -11π/6-11π/61
Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1
Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3
Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3 SOHCAHTOA
Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3 SOHCAHTOAGiven arighttriangle and one of the small angles, say A, the adjacent and the opposite of the angle A are as shown:
SOHCAHTOAIf the angle A is placed in the standard position, hypotenuseoppositeAadjacent
SOHCAHTOAIf the angle A is placed in the standard position, then the trig-values of A are:opposite O Sin(A) = = hypotenuse H hypotenuseoppositeadjacent A ACos(A) = = adjacenthypotenuse H opposite O = Tan(A) = adjacent A
SOHCAHTOAIf the angle A is placed in the standard position, then the trig-values of A are:opposite O Sin(A) = = hypotenuse H hypotenuseoppositeadjacent A ACos(A) = = adjacenthypotenuse H opposite O = Tan(A) = adjacent A One checks easily that these trig-values are the same as the ones defined via the unit circle.
SOHCAHTOAIf the angle A is placed in the standard position, then the trig-values of A are:opposite O Sin(A) = = hypotenuse H hypotenuseoppositeadjacent A ACos(A) = = adjacenthypotenuse H opposite O = Tan(A) = adjacent A One checks easily that these trig-values are the same as the ones defined via the unit circle. Hence we use "SOHCAHTOA" to remember the definition of trig-functions for positive angles that are smaller than 90o.
SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)a
SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first,
SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first, we'vea2 + 82 = 112
SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first, we'vea2 + 82 = 112a2 = 121 – 64 = 57a = 57
SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first, we'vea2 + 82 = 112a2 = 121 – 64 = 57a = 57Opp 57 Hence tan(A) == Adj 8 8 Opp = sin(B) =11 Hyp

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t1 angles and trigonometric functions

  • 2. Angular MeasurementsThere are two systems of angular measurements
  • 3. Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system
  • 4. Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o.
  • 5. Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'.
  • 6. Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1".
  • 7. Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1". The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering.
  • 8. Angular MeasurementsThere are two systems of angular measurements.I. The degree system II. The radian system Divide a pizza into 360 equal slices, the tip of a slice is an 1 degree angle noted as 1o. One degree is divided into 60 minutes, each minute is denoted as 1'. One minute is divided into 60 seconds, each second is denoted as 1". The degree system is the same as our time-system of hour-minute-second and its used mostly in science and engineering. In mathematics, the radian system is used more often because it's relationship with the geometry of circles.
  • 9. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.
  • 10. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.r = 1
  • 11. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Its given by the equation x2 + y2 = 1.r = 1
  • 12. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1
  • 13. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1
  • 14. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.
  • 15. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems
  • 16. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems180o = π rad
  • 17. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems180o = π rad 1o = rad π180
  • 18. Radian MeasurementsThe unitcircle is the circle centered at (0, 0) with radius 1.Arc length as anglemeasurement for Its given by the equation x2 + y2 = 1.The radianmeasurement of an angle is the length of the arc that the angle cuts out on the unitcircle.r = 1Hence 2π(the circumference of the unit circle) is the radian measurement corresponds to the 360o angle.The following formulas convert the measurements between Degree and Radian systems180o = π rad 1o = rad = 1 rad  57oπ180oπ180
  • 19. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position.
  • 20. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.
  • 21. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is –
  • 22. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), (x , y)(1,0)
  • 23. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)(1,0)yx
  • 24. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)y=sin()(1,0)x=cos()yx
  • 25. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)y=sin()(1,0)x=cos()yxNote: tan() = slope of the dial
  • 26. Definition of Trigonometric FunctionsWhen an angle is formed by dialing a dial against the positive x-axis, we say the angle is in the standard position. If the angle is dialed counter clockwisely, it's measurement is set to be positive. If the angle is dialed clockwisely, it's negative.  is + is – Given an angle  in the standard position, let the coordinate of the tip of the dial on the unit circle be (x , y), we define that: cos() = x,sin() = y, tan() = (x , y)tan()y=sin()(1,0)x=cos()yxNote: tan() = slope of the dial
  • 27. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4
  • 28. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. π/4
  • 29. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. π/4π/4π/4
  • 30. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, π/4π/4π/4
  • 31. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2π/4π/4π/4
  • 32. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)
  • 33. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. π/6
  • 34. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. π/6π/6c c = 2aπ/3π/3
  • 35. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. π/6π/6c c = 2aπ/3π/3
  • 36. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2π/6π/6c c = 2aπ/3π/3
  • 37. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 a2 + b2 = 4a2 b2 = 3a2π/6π/6c c = 2aπ/3π/3
  • 38. Two Important Right TrianglesThere are two important classes of right triangles, the rt-triangles, and the rt-triangles.π/6π/4A rt-triangle is an isosceles triangle, so the legs are equal, say it’s a. Let the hypotenuse be c then a2 + a2 = c2, 2a2 = c2 2a2 = c2 or a2 = cπ/4π/4π/4(21.414)For the rt-triangles, take two of them and put them back to back to form an equilateral triangle as shown. Hence c = a + a or c = 2a. So a2 + b2 = (2a)2 a2 + b2 = 4a2 b2 = 3a2 b = 3a2 or b = a3π/6π/6c c = 2aπ/3π/3
  • 39. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .
  • 40. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/412/2π/42/2
  • 41. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2
  • 42. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric Values
  • 43. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric ValuesThe trig-values of angles depend on the positions of the angle.
  • 44. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric ValuesThe trig-values of angles depend on the positions of the angle. Following are the important positions of the dials for which we may find the exact trig-values.
  • 45. Two Important Right TrianglesFrom the above, we obtain the following two triangles that are useful for extracting the trigonometric values of angles related to π/4, π/6 and π/6 .π/4π/3112/21/2π/6π/43/22/2Important Trigonometric ValuesThe trig-values of angles depend on the positions of the angle. Following are the important positions of the dials for which we may find the exact trig-values. Note that  and  + 2nπhave the same position where n is an integer.
  • 46. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.
  • 47. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.0, ±2π, ±4π..
  • 48. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..
  • 49. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.Kπ2
  • 50. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2
  • 51. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..
  • 52. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..Angles with measurements of rad are diagonals.Kπ Frank Ma20064
  • 53. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma20064
  • 54. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ4
  • 55. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200645π/4, -3π/4..
  • 56. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..
  • 57. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..Angles with measurements of (reduced) orrad.KπKπ63
  • 58. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..Angles with measurements of (reduced) orrad.π/6, -11π/6..KπKπ63
  • 59. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..Angles with measurements of (reduced) orrad.π/6, -11π/6..KπKπ63
  • 60. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ63
  • 61. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ7π/6,..634π/3,..
  • 62. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ7π/6,..6311π/6,..4π/3,..5π/3,..
  • 63. Important Trigonometric ValuesAngles with measurements of Nπrad, where N is an integer, correspond to the x-axial angles.±π, ±3π..0, ±2π, ±4π..Angles with measurements of rad correspondto the y-axial angles.π/2, 5π/2..Kπ2-π/2, 3π/2..π/4, -7π/4..3π/4, -5π/4..Angles with measurements of rad are diagonals.Kπ Frank Ma200647π/4, -π/4..5π/4, -3π/4..π/3, -5π/3..2π/3,..Angles with measurements of (reduced) orrad.5π/6,..π/6, -11π/6..KπKπ7π/6,..6311π/6,..4π/3,..5π/3,..
  • 64. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3π
  • 65. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3π-3π(-1, 0)
  • 66. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0
  • 67. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4
  • 68. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/45π/4
  • 69. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,5π/4
  • 70. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,15π/4
  • 71. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,we get the coordinate = (-2/2, -2/2).15π/4(-2/2, -2/2)
  • 72. Important Trigonometric ValuesExample A: Draw the angle, label the coordinates of the corresponding position on the unit circle and list the sine, cosine, and tangent trig-values.a.  = -3πsin(-3π) = 0 cos(-3π) = -1 -3π(-1, 0) tan(-3π) = 0 b.  = 5π/4Place the π/4-rt-triangle as shown,we get the coordinate = (-2/2, -2/2).sin(5π/4) = -2/2 cos(5π/4) = -2/2 15π/4(-2/2, -2/2) tan(5π/4) = 1
  • 74. Important Trigonometric Valuesc.  = -11π/6-11π/61
  • 75. Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1
  • 76. Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3
  • 77. Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3 SOHCAHTOA
  • 78. Important Trigonometric Valuesc.  = -11π/6Place the π/6-rt-triangle as shown, we get the coordinate = (3/2, 1/2).-11π/6(3/2, ½) 1sin(-11π/6) = 1/2 cos(-11π/6) = 3/2 tan(-11π/6) = 1/3 =3/3 SOHCAHTOAGiven arighttriangle and one of the small angles, say A, the adjacent and the opposite of the angle A are as shown:
  • 79. SOHCAHTOAIf the angle A is placed in the standard position, hypotenuseoppositeAadjacent
  • 80. SOHCAHTOAIf the angle A is placed in the standard position, then the trig-values of A are:opposite O Sin(A) = = hypotenuse H hypotenuseoppositeadjacent A ACos(A) = = adjacenthypotenuse H opposite O = Tan(A) = adjacent A
  • 81. SOHCAHTOAIf the angle A is placed in the standard position, then the trig-values of A are:opposite O Sin(A) = = hypotenuse H hypotenuseoppositeadjacent A ACos(A) = = adjacenthypotenuse H opposite O = Tan(A) = adjacent A One checks easily that these trig-values are the same as the ones defined via the unit circle.
  • 82. SOHCAHTOAIf the angle A is placed in the standard position, then the trig-values of A are:opposite O Sin(A) = = hypotenuse H hypotenuseoppositeadjacent A ACos(A) = = adjacenthypotenuse H opposite O = Tan(A) = adjacent A One checks easily that these trig-values are the same as the ones defined via the unit circle. Hence we use "SOHCAHTOA" to remember the definition of trig-functions for positive angles that are smaller than 90o.
  • 83. SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)a
  • 84. SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first,
  • 85. SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first, we'vea2 + 82 = 112
  • 86. SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first, we'vea2 + 82 = 112a2 = 121 – 64 = 57a = 57
  • 87. SOHCAHTOAExample B: Given the rt-triangle, find tan(A) and sin(B)aTo find a first, we'vea2 + 82 = 112a2 = 121 – 64 = 57a = 57Opp 57 Hence tan(A) == Adj 8 8 Opp = sin(B) =11 Hyp