Math resources trigonometric_formulas
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This document provides formulas and definitions for trigonometric functions including the definitions of sine, cosine, and tangent using right triangles and the unit circle. It also includes information on domains, ranges, periods, identities, inverse trig functions, complex numbers, conic sections, and formulas for working with angles in degrees and radians. Key aspects covered are the definitions of trig functions, trig identities, inverse trig functions, and formulas for circles, ellipses, hyperbolas, and parabolas.
![Identities and Formulas
Tangent and Cotangent Identities
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
Reciprocal Identities
sin θ =
1
csc θ
csc θ =
1
sin θ
cos θ =
1
sec θ
sec θ =
1
cos θ
tan θ =
1
cot θ
cot θ =
1
tan θ
Pythagorean Identities
sin2
θ + cos2
θ = 1
tan2
θ + 1 = sec2
θ
1 + cot2
θ = csc2
θ
Even and Odd Formulas
sin(−θ) = − sin θ
cos(−θ) = cos θ
tan(−θ) = − tan θ
csc(−θ) = − csc θ
sec(−θ) = sec θ
cot(−θ) = − cot θ
Periodic Formulas
If n is an integer
sin(θ + 2πn) = sin θ
cos(θ + 2πn) = cos θ
tan(θ + πn) = tan θ
csc(θ + 2πn) = csc θ
sec(θ + 2πn) = sec θ
cot(θ + πn) = cot θ
Double Angle Formulas
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos2
θ − sin2
θ
= 2 cos2
θ − 1
= 1 − 2 sin2
θ
tan(2θ) =
2 tan θ
1 − tan2
θ
Degrees to Radians Formulas
If x is an angle in degrees and t is an angle in
radians then:
π
180◦
=
t
x
⇒ t =
πx
180◦
and x =
180◦
t
π
Half Angle Formulas
sin θ = ±
1 − cos(2θ)
2
cos θ = ±
1 + cos(2θ)
2
tan θ = ±
1 − cos(2θ)
1 + cos(2θ)
Sum and Difference Formulas
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β sin α sin β
tan(α ± β) =
tan α ± tan β
1 tan α tan β
Product to Sum Formulas
sin α sin β =
1
2
[cos(α − β) − cos(α + β)]
cos α cos β =
1
2
[cos(α − β) + cos(α + β)]
sin α cos β =
1
2
[sin(α + β) + sin(α − β)]
cos α sin β =
1
2
[sin(α + β) − sin(α − β)]
Sum to Product Formulas
sin α + sin β = 2 sin
α + β
2
cos
α − β
2
sin α − sin β = 2 cos
α + β
2
sin
α − β
2
cos α + cos β = 2 cos
α + β
2
cos
α − β
2
cos α − cos β = −2 sin
α + β
2
sin
α − β
2
Cofunction Formulas
sin
π
2
− θ = cos θ
csc
π
2
− θ = sec θ
tan
π
2
− θ = cot θ
cos
π
2
− θ = sin θ
sec
π
2
− θ = csc θ
cot
π
2
− θ = tan θ
2](https://image.slidesharecdn.com/mathresourcestrigonometricformulas-151109183541-lva1-app6892/85/Math-resources-trigonometric_formulas-2-320.jpg)
Math resources trigonometric_formulas
- 1. Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < π 2 or 0◦ < θ < 90◦ hypotenuse adjacent opposite θ sin θ = opp hyp csc θ = hyp opp cos θ = adj hyp sec θ = hyp adj tan θ = opp adj cot θ = adj opp Unit Circle Definition Assume θ can be any angle. x y y x 1 (x, y) θ sin θ = y 1 csc θ = 1 y cos θ = x 1 sec θ = 1 x tan θ = y x cot θ = x y Domains of the Trig Functions sin θ, ∀ θ ∈ (−∞, ∞) cos θ, ∀ θ ∈ (−∞, ∞) tan θ, ∀ θ = n + 1 2 π, where n ∈ Z csc θ, ∀ θ = nπ, where n ∈ Z sec θ, ∀ θ = n + 1 2 π, where n ∈ Z cot θ, ∀ θ = nπ, where n ∈ Z Ranges of the Trig Functions −1 ≤ sin θ ≤ 1 −1 ≤ cos θ ≤ 1 −∞ ≤ tan θ ≤ ∞ csc θ ≥ 1 and csc θ ≤ −1 sec θ ≥ 1 and sec θ ≤ −1 −∞ ≤ cot θ ≤ ∞ Periods of the Trig Functions The period of a function is the number, T, such that f (θ +T ) = f (θ ) . So, if ω is a fixed number and θ is any angle we have the following periods. sin(ωθ) ⇒ T = 2π ω cos(ωθ) ⇒ T = 2π ω tan(ωθ) ⇒ T = π ω csc(ωθ) ⇒ T = 2π ω sec(ωθ) ⇒ T = 2π ω cot(ωθ) ⇒ T = π ω 1
- 2. Identities and Formulas Tangent and Cotangent Identities tan θ = sin θ cos θ cot θ = cos θ sin θ Reciprocal Identities sin θ = 1 csc θ csc θ = 1 sin θ cos θ = 1 sec θ sec θ = 1 cos θ tan θ = 1 cot θ cot θ = 1 tan θ Pythagorean Identities sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ Even and Odd Formulas sin(−θ) = − sin θ cos(−θ) = cos θ tan(−θ) = − tan θ csc(−θ) = − csc θ sec(−θ) = sec θ cot(−θ) = − cot θ Periodic Formulas If n is an integer sin(θ + 2πn) = sin θ cos(θ + 2πn) = cos θ tan(θ + πn) = tan θ csc(θ + 2πn) = csc θ sec(θ + 2πn) = sec θ cot(θ + πn) = cot θ Double Angle Formulas sin(2θ) = 2 sin θ cos θ cos(2θ) = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ tan(2θ) = 2 tan θ 1 − tan2 θ Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then: π 180◦ = t x ⇒ t = πx 180◦ and x = 180◦ t π Half Angle Formulas sin θ = ± 1 − cos(2θ) 2 cos θ = ± 1 + cos(2θ) 2 tan θ = ± 1 − cos(2θ) 1 + cos(2θ) Sum and Difference Formulas sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β tan(α ± β) = tan α ± tan β 1 tan α tan β Product to Sum Formulas sin α sin β = 1 2 [cos(α − β) − cos(α + β)] cos α cos β = 1 2 [cos(α − β) + cos(α + β)] sin α cos β = 1 2 [sin(α + β) + sin(α − β)] cos α sin β = 1 2 [sin(α + β) − sin(α − β)] Sum to Product Formulas sin α + sin β = 2 sin α + β 2 cos α − β 2 sin α − sin β = 2 cos α + β 2 sin α − β 2 cos α + cos β = 2 cos α + β 2 cos α − β 2 cos α − cos β = −2 sin α + β 2 sin α − β 2 Cofunction Formulas sin π 2 − θ = cos θ csc π 2 − θ = sec θ tan π 2 − θ = cot θ cos π 2 − θ = sin θ sec π 2 − θ = csc θ cot π 2 − θ = tan θ 2
- 3. Unit Circle 0◦ , 2π (1, 0) 180◦ , π (−1, 0) (0, 1) 90◦ , π 2 (0, −1) 270◦ , 3π 2 30◦ , π 6 ( √ 3 2 , 1 2 )45◦ , π 4 ( √ 2 2 , √ 2 2 ) 60◦ , π 3 (1 2 , √ 3 2 ) 120◦ , 2π 3 (−1 2 , √ 3 2 ) 135◦ , 3π 4 (− √ 2 2 , √ 2 2 ) 150◦ , 5π 6 (− √ 3 2 , 1 2 ) 210◦ , 7π 6 (− √ 3 2 , −1 2 ) 225◦ , 5π 4 (− √ 2 2 , − √ 2 2 ) 240◦ , 4π 3 (−1 2 , − √ 3 2 ) 300◦ , 5π 3 (1 2 , − √ 3 2 ) 315◦ , 7π 4 ( √ 2 2 , − √ 2 2 ) 330◦ , 11π 6 ( √ 3 2 , −1 2 ) For any ordered pair on the unit circle (x, y) : cos θ = x and sin θ = y Example cos (7π 6 ) = − √ 3 2 sin (7π 6 ) = −1 2 3
- 4. Inverse Trig Functions Definition θ = sin−1 (x) is equivalent to x = sin θ θ = cos−1 (x) is equivalent to x = cos θ θ = tan−1 (x) is equivalent to x = tan θ Domain and Range Function θ = sin−1 (x) θ = cos−1 (x) θ = tan−1 (x) Domain −1 ≤ x ≤ 1 −1 ≤ x ≤ 1 −∞ ≤ x ≤ ∞ Range − π 2 ≤ θ ≤ π 2 0 ≤ θ ≤ π − π 2 < θ < π 2 Inverse Properties These properties hold for x in the domain and θ in the range sin(sin−1 (x)) = x cos(cos−1 (x)) = x tan(tan−1 (x)) = x sin−1 (sin(θ)) = θ cos−1 (cos(θ)) = θ tan−1 (tan(θ)) = θ Other Notations sin−1 (x) = arcsin(x) cos−1 (x) = arccos(x) tan−1 (x) = arctan(x) Law of Sines, Cosines, and Tangents a b c α β γ Law of Sines sin α a = sin β b = sin γ c Law of Cosines a2 = b2 + c2 − 2bc cos α b2 = a2 + c2 − 2ac cos β c2 = a2 + b2 − 2ab cos γ Law of Tangents a − b a + b = tan 1 2 (α − β) tan 1 2 (α + β) b − c b + c = tan 1 2 (β − γ) tan 1 2 (β + γ) a − c a + c = tan 1 2 (α − γ) tan 1 2 (α + γ) 4
- 5. Complex Numbers i = √ −1 i2 = −1 i3 = −i i4 = 1 √ −a = i √ a, a ≥ 0 (a + bi) + (c + di) = a + c + (b + d)i (a + bi) − (c + di) = a − c + (b − d)i (a + bi)(c + di) = ac − bd + (ad + bc)i (a + bi)(a − bi) = a2 + b2 |a + bi| = √ a2 + b2 Complex Modulus (a + bi) = a − bi Complex Conjugate (a + bi)(a + bi) = |a + bi|2 DeMoivre’s Theorem Let z = r(cos θ + i sin θ), and let n be a positive integer. Then: zn = rn (cos nθ + i sin nθ). Example: Let z = 1 − i, find z6 . Solution: First write z in polar form. r = (1)2 + (−1)2 = √ 2 θ = arg(z) = tan−1 −1 1 = − π 4 Polar Form: z = √ 2 cos − π 4 + i sin − π 4 Applying DeMoivre’s Theorem gives : z6 = √ 2 6 cos 6 · − π 4 + i sin 6 · − π 4 = 23 cos − 3π 2 + i sin − 3π 2 = 8(0 + i(1)) = 8i 5
- 6. Finding the nth roots of a number using DeMoivre’s Theorem Example: Find all the complex fourth roots of 4. That is, find all the complex solutions of x4 = 4. We are asked to find all complex fourth roots of 4. These are all the solutions (including the complex values) of the equation x4 = 4. For any positive integer n , a nonzero complex number z has exactly n distinct nth roots. More specifically, if z is written in the trigonometric form r(cos θ + i sin θ), the nth roots of z are given by the following formula. (∗) r 1 n cos θ n + 360◦ k n + i sin θ n + 360◦ k n , for k = 0, 1, 2, ..., n − 1. Remember from the previous example we need to write 4 in trigonometric form by using: r = (a)2 + (b)2 and θ = arg(z) = tan−1 b a . So we have the complex number a + ib = 4 + i0. Therefore a = 4 and b = 0 So r = (4)2 + (0)2 = 4 and θ = arg(z) = tan−1 0 4 = 0 Finally our trigonometric form is 4 = 4(cos 0◦ + i sin 0◦ ) Using the formula (∗) above with n = 4, we can find the fourth roots of 4(cos 0◦ + i sin 0◦ ) • For k = 0, 4 1 4 cos 0◦ 4 + 360◦ ∗ 0 4 + i sin 0◦ 4 + 360◦ ∗ 0 4 = √ 2 (cos(0◦ ) + i sin(0◦ )) = √ 2 • For k = 1, 4 1 4 cos 0◦ 4 + 360◦ ∗ 1 4 + i sin 0◦ 4 + 360◦ ∗ 1 4 = √ 2 (cos(90◦ ) + i sin(90◦ )) = √ 2i • For k = 2, 4 1 4 cos 0◦ 4 + 360◦ ∗ 2 4 + i sin 0◦ 4 + 360◦ ∗ 2 4 = √ 2 (cos(180◦ ) + i sin(180◦ )) = − √ 2 • For k = 3, 4 1 4 cos 0◦ 4 + 360◦ ∗ 3 4 + i sin 0◦ 4 + 360◦ ∗ 3 4 = √ 2 (cos(270◦ ) + i sin(270◦ )) = − √ 2i Thus all of the complex roots of x4 = 4 are: √ 2, √ 2i, − √ 2, − √ 2i . 6
- 7. Formulas for the Conic Sections Circle StandardForm : (x − h)2 + (y − k)2 = r2 Where (h, k) = center and r = radius Ellipse Standard Form for Horizontal Major Axis : (x − h)2 a2 + (y − k)2 b2 = 1 Standard Form for V ertical Major Axis : (x − h)2 b2 + (y − k)2 a2 = 1 Where (h, k)= center 2a=length of major axis 2b=length of minor axis (0 < b < a) Foci can be found by using c2 = a2 − b2 Where c= foci length 7
- 8. More Conic Sections Hyperbola Standard Form for Horizontal Transverse Axis : (x − h)2 a2 − (y − k)2 b2 = 1 Standard Form for V ertical Transverse Axis : (y − k)2 a2 − (x − h)2 b2 = 1 Where (h, k)= center a=distance between center and either vertex Foci can be found by using b2 = c2 − a2 Where c is the distance between center and either focus. (b > 0) Parabola Vertical axis: y = a(x − h)2 + k Horizontal axis: x = a(y − k)2 + h Where (h, k)= vertex a=scaling factor 8
- 9. x Example : sin 5π 4 = − √ 2 2 f(x) f(x) = sin(x) 0 π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6 π 7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6 2π 1 -1 1 2 √ 2 2 √ 3 2 −1 2 − √ 2 2 − √ 3 2 x Example : cos 7π 6 = − √ 3 2 f(x) f(x) = cos(x) 0 π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6 π 7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6 2π 1 -1 1 2 √ 2 2 √ 3 2 −1 2 − √ 2 2 − √ 3 2 9
- 10. x f(x) f(x) = tan x π 2−π 2 √ 3 3 1 √ 3 − √ 3 3 −1 − √ 3 π 4−π 4 0 π 6−π 6 π 3−π 3 2π 3−2π 3 3π 4−3π 4 5π 6−5π 6 π−π 10