5.5 Multiple-Angle Formulas

Students will use multiple-angle formulas to rewrite and evaluate trigonometric functions.

Students will use power-reducing formulas to rewrite and evaluate trigonometric functions.

Students will use half-angle formulas to rewrite and evaluate trigonometric functions.

Students will use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions.

Section 5.5, Double-Angle Formulas, pg. 375

Why is ?

Remember:

sin sin cos2 2 sin( ) sin cos cos sina b a b a b

Example 1

2 2 0cos sin Solve.

Try #10 on p. 382

Why is ?

Remember:

cos cos sin2 2 2 cos( ) cos cos sin sina b a b a b

Example 3Use the following to find and : sin 2 cos2 tan 2

sin 2

cos2

tan 2

cos 5

133

22

Try #18 on p. 382

Section 5.5, Half-Angle Formulas, pg. 378

Use the figure (p.382 #33-40) to find the exact value of the trigonometric function.

cos2

Try #34 on p. 382

Example 6

Find the exact value of sin105

Try #42 on p. 383

Find the following values given in quad. II sin

2

cos2

tan2

sin 5

13

Try #50 on p. 383

Use the half-angle formulas to simplify the expression.

1 6

2

cos x

Try #56 on p. 383

Example 7

Find all solutions if in the interval

2 22

2 2 sin cos [ , )0 2

Section 5.5, Product-to-Sum Formulas, pg. 379

Example 8 Rewrite the product as a sum or differencecos sin5 4

Section 5.5, Sum-to-Product Formulas, pg. 380

Example 9

Find the exact value of cos cos195 105

Example 10

Find all solutions of in the interval sin sin5 3 0x x [ , )0 2