1 Start Up Day 38 1.Solve over the interval 2. Solve:

2 Sum and Difference Formulas OBJECTIVE : SWBAT demonstrate and understanding for uses of the sum and difference, half-angle, and double-angle formulas for sine, cosine, and tangent. EQ : How can the sum and difference formulas determine exact values, verify identities and help solve trigonometric equations? CLASS WORK: p. 425 #3, 4, 11, 15, 27, 47 &53 HOMELEARNING: p.425 ‐ 427 # 8, 12, 13, 16, 22, 28, 34, 40, 48, 54

3 Sum and Difference Formulas In this section, we will study the uses of several trigonometric identities and formulas. (Notice that you have the condensed version on your “shower” sheet.) 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.

4 Example 1 – Evaluating a Trigonometric Function Find the exact value of. Solution: To find the exact value of, use the fact that ( Sometimes it is helpful to convert to degrees. Think of 15°…then think of two special angles from the unit circle that either add or subtract to build 15°…60°-45° will work! Consequently, the formula for sin(u – v) yields (You may also choose to write this in degree form)

5 Example 1 – Solution Expand by applying the formula Replace with the exact Unit Circle values Multiply & re-write with The common denominator. Try checking this result on your calculator. You will find that  cont’d

6 Using Sum and Difference Formulas Example 2 shows how to use a difference formula to prove OR VERIFY the co-function identity = sin x.

7 Example 2 – Proving a Cofunction identity Solution: Using the formula for cos(u – v), you have Expand Replace Values Simplify Prove the co-function identity = sin x.

8 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.

9 Example 3 – Solving a Trigonometric Equation in the interval. Solution: Using sum and difference formulas, expand the equation as Combine like terms Find all solutions of

10 Example 3 – Solution Replace with exact values Simplify, if possible Divide Identify the angles on the UNIT Circle….. So, the only solutions in the interval are Finally—Remember that you can check by graphing! cont’d and.