Trigonometric functions - PreCalculus
- 2. Our Friend the Unit Circle!hypradius=1oppƟadjThis circle will be used for everything in this section. It helps us with all the functions, ratios, and calculations we will learn. This is why it is our new best friend.
- 3. Using the Unit CircleTrigonometrysin Ɵ =
- 4. cos Ɵ =
- 5. tan Ɵ = =
- 6. csc Ɵ = =
- 7. sec Ɵ = =
- 8. cot Ɵ = = opphypadjhypsin Ɵcos Ɵoppadjhypopp 1 1sin Ɵ 1 1cos Ɵhypadjcos Ɵsin Ɵadjoppcsc=cosecant
- 9. sec=secant
- 10. cot=cotangentGraphing Using the Unit Circle 90°180° 0/360° 240°Steps to Graphing on the Unit CircleWhen given an angle (ex. 135°), draw a curve from 0° to the given angle.
- 11. Draw a line connecting the curve you just drew to the edge of the circle.Draw a dashed line from the edge of the circle to make a right triangle.Calculate the angle measure inside the triangle (180-135 = 45)Find the sine, cosine, and tangent of the angle (you may use a calculator for this if you wish). 90°180° 0/360° 240°45°
- 12. TrigonometryUsing the triangle at right, solve for sin Ɵ, cosƟ, and tanƟ.opphypsin Ɵ = = 3/4cosƟ = = 3/5 tan Ɵ = = 4/3adjhyp5oppadj4Ɵ3
- 13. Using the triangle at right, solve for cscƟ, sec Ɵ, and cot Ɵ.hypoppcscƟ = = 5/4secƟ = = 5/3cotƟ = = 3/4hypadj54adjoppƟ3
- 14. Radian MeasureRadians use π ratios instead of degrees and are fractions, not whole numbers , π,2π,etc…π , π , π ,π, 3π, 5π6 4 3 2 4 6
- 15. Degree MeasureDegrees do not use π ratios and are whole numbers instead of fractions15°, 30°, 45°, 60°, 90°, 120°, 180°, 360°, etc… 90°180° 0/360° 240°
- 16. Converting between Radians and DegreesWhen converting from Radians to Degrees:Multiply the radian ratio by 180Divide the radian ratio by πThe two π symbols will cancel each other out and all that will be left is a simple mathematical equationExample: = = 150°5π6180π5(180) 6
- 17. Converting Between Degrees and RadiansWhen converting from Degrees to Radians:Multiply the degree measure by πDivide the degree measure by 180Example:120° π = 120π= 2π___180____ 180__ 3
- 18. Arc Length of a Given AngleThe equation for finding the arc length of a given angle is:s = dπƟ s = arc lengthd = diameter (2r)Ɵ = given angle____ 360sƟr
- 19. Arc Length of an Angle in a Given CircleFind the arc length of 60° in a circle with radius = 5.s = dπƟ 360s = 10π60 360s = 600π 360s = 5π 3s = 5.236r=560°
- 20. CitationsInformation:http://www.math-prof.com/Geom/Geom_Ch_32.aspAll other information is from my previous pre-calculus experience. I am a math major, so I should know all of this anyway.
- 21. Citations cont…Pictures:Title slide pictures, photos on slides 10 & 11, and teacher photo on Information Citation slide from PowerPoint Clip ArtUnit Circle and Triangles made by me using PowerPointPi Pie photo on slide 10 from: pauladamsmith, 13 January 2008 via Flickr, Creative Commons Attribution.
- 22. Citations cont…Lesson Plan from: http://www.michigan.gov/documents/ PreCalc_167750_7.pdfLesson P6.1 – “Define (using the unit circle), graph, and use all trigonometric functions of any angle. Convert between radian and degree measure. Calculate arc lengths in given circles.”