MATH 10560: CALCULUS II
TRIGONOMETRIC FORMULAS
Basic Identities
The functions cos(θ) and sin(θ) are defined to be the xand ycoordinates of the point at an angle of θ
on the unit circle. Therefore, sin(−θ) = −sin(θ), cos(−θ) = cos(θ), and sin2(θ) + cos2(θ) = 1. The other
trigonometric functions are defined in terms of sine and cosine:
tan(θ) = sin(θ)/cos(θ) cot(θ) = cos(θ)/sin(θ) = 1/tan(θ)
sec(θ) = 1/cos(θ) csc(θ) = 1/sin(θ)
Dividing sin2(θ) + cos2(θ) = 1 by cos2(θ) or sin2(θ) gives tan2(θ) + 1 = sec2(θ) and 1 + cot2(θ) = csc2(θ).
Addition Formulas
The following two addition formulas are fundamental:
sin(A+B) = sin(A) cos(B) + cos(A) sin(B)
cos(A+B) = cos(A) cos(B)−sin(A) sin(B)
They can be used to prove simple identities like sin(π/2−θ) = sin(π/2) cos(θ) + cos(π/2) sin(θ) = cos(θ), or
cos(x−π) = cos(x) cos(π)−sin(x) sin(π) = −cos(x). If we set A=Bin the addition formulas we get the
double-angle formulas:
sin(2A) = 2 sin(A) cos(A) cos(2A) = cos2(A)−sin2(A)
The formula for cos(2A) is often rewritten by replacing cos2(A) with 1 −sin2(A) or replacing sin2(A) with
1−cos2(A) to get
cos(2A) = 1 −2 sin2(A) cos(2A) = 2 cos2(A)−1
Solving for sin2(A) and cos2(A) yields identities important for integration:
sin2(A) = 1
2(1 −cos(2A)) cos2(A) = 1
2(1 + cos(2A))
The addition formulas can also be combined to give other formulas important for integration:
sin Asin B=1
2[cos(A−B)−cos(A+B)]
cos Acos B=1
2[cos(A−B) + cos(A+B)]
sin Acos B=1
2[sin(A−B) + sin(A+B)]
Derivatives and Integrals
sin′(x) = cos(x) sec′(x) = sec(x) tan(x)
cos′(x) = −sin(x) csc′(x) = −csc(x) cot(x)
tan′(x) = sec2(x) cot′(x) = −csc2(x)
Rsin(x)dx =−cos(x) + CRsec(x)dx = ln |sec(x) + tan(x)|+C
Rcos(x)dx = sin(x) + CRcsc(x)dx = ln |csc(x)−cot(x)|+C
Rtan(x)dx = ln |sec(x)|+CRcot(x)dx =−ln |csc(x)|+C
