1. 2 Analytic Trigonometry
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2.4
SUM AND DIFFERENCE FORMULAS
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3. What You Should Learn
Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations.

4. Using Sum and Difference Formulas

5. Using Sum and Difference Formulas
In this section, we will study the uses of several trigonometric identities and formulas.
Example 1 shows how sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles.

6. Example 1 – Evaluating a Trigonometric Function
Find the exact value of
Solution:
To find the exact value of , use the fact that
Consequently, the formula for sin(u – v) yields

7. Example 1 – Solution
cont’d
Try checking this result on your calculator. You will find that 

8. Using Sum and Difference Formulas
Example 5 shows how to use a difference formula to prove the cofunction identity
= sin x.

9. Example 5 – Proving a Cofunction identity
Solution:
Using the formula for cos(u – v), you have
Prove the cofunction identity
= sin x.

10. Using Sum and Difference Formulas
Sum and difference formulas can be used to rewrite expressions such as
and , where n is an integer as expressions involving only sin  or cos  .
The resulting formulas are called reduction formulas.

11. Example 7 – Solving a Trigonometric Equation
in the interval
Solution:
Using sum and difference formulas, rewrite the equation as
Find all solutions of

12. Example 7 – Solution So, the only solutions in the interval are and .
cont’d
So, the only solutions in the interval are
and .