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Pre – Calculus
Unit 4
Section 5.4: Sum and Difference Identities
Objectives:
Use Sum and Difference Identities to
- evaluate trigonometric functions
- solve trigonometric equations
Sum and Difference Identities
These are identities involving 2 variables.
Alpha () and Beta () represent angle measures.
These identities are used to find exact values of trig functions of angles that are LESS COMMON
Example 1: Find the exact value of
a) cos 75°
b) tan
11𝜋
12
c) sin15
Example 2:
a) An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t, where 255 is a degree
measure. Rewrite the formula in terms of the sum of two angle measures.
b) An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t. Use a sum identity to find
the exact current after 1 second.
Example 3: Rewrite as a single trigonometric function
a) Find the exact value of
tan 78°−tan 18°
c) Find the exact value of sin
7𝜋
6
cos
5𝜋
6
−cos
𝜋
𝜋
𝜋
𝜋
3
4
3
4
b) Simplify sin cos +cos sin
1 + tan 78° tan 18°
7𝜋
6
sin
5𝜋
6
𝜋
𝜋
2
2
d) sin sin 𝑥 + cos cos 𝑥
Example 4: Write as an algebraic expression that does not involve trigonometric functions.
a) cos (arcsin
√3
2
+ arccos 𝑥)
b) sin(arccos 2x + arcsin x)
Example 5: Verify:
a) cos (–θ) = cos θ
b) tan (– ) = –tan 
Reduction Identity
𝜋
Sum and Difference identities can be used to rewrite trigonometric expressions in which one of the angles is a multiple of 90 or
2
radians.
The resulting expression is called a reduction identitity because it reduces the complexity of the expression.
Example 6: Verfiy
𝜋
b) tan (x – 360°) = tan x
a) cos (𝜃 − ) = sin 𝜃
2
𝜋
c) sin (𝜃 − ) = − cos 𝜃
2
Example 7: Find the solutions of the equation on the interval [0, 2).
𝜋
𝜋
4
4
a) sin (𝑥 + ) − sin (𝑥 − ) = 0
𝜋
𝜋
4
4
b) cos (𝑥 − ) + sin (𝑥 + ) = 0