Chapter 10 Trigonometric functions - Ugrad.math.ubc.ca

Math 102 Notes Chapter 10
b
θ
c
a
Figure 10.9:
Angle sum identities
The trigonometric functions are nonlinear. This means that, for example, the sine of the sum of
two angles is not just the sum of the two sines. One can use the law of cosines and other geometric
ideas to establish the following two relationships:
sin(A + B) = sin(A) cos(B) + sin(B) cos(A)
cos(A + B) = cos(A) cos(B) − sin(A) sin(B)
These two identities appear in many calculations, and will be important for computing derivatives
of the basic trigonometric formulae.
Related identities
The identities for the sum of angles can be used to derive a number of related formulae. For example,
by replacing B by −B we get the angle difference identities:
sin(A − B) = sin(A) cos(B) − sin(B) cos(A)
cos(A − B) = cos(A) cos(B) + sin(A) sin(B)
By setting θ = A = B in these we find the subsidiary double angle formulae:
and these can also be written in the form
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos 2 (θ) − sin 2 (θ)
2 cos 2 (θ) = 1 + cos(2θ)
2 sin 2 (θ) = 1 − cos(2θ).
(The latter four are quite useful in integration methods.)
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