1. Power-reducing & Half angle identities
5-4 cont.
Power-reducing & Half angle identities

2. Power-reducing identities
Use the double angle identities to come up with an identity for sin2x, cos2x, and tan2x.

3. Power-reducing Identities

4. Deriving Half-Angle Formulas Using Sine of a Double Angle
Recall that cos 2q = 1- 2sin2q
Substituting in q/2 for q, you obtain
cos 2(q/2) = 1- 2sin2(q/2)
Simplifying and solving for sin (q/2)…

5. Deriving Half-Angle Formulas Using Cosine of a Double Angle
Half-Angle Identities
Deriving Half-Angle Formulas Using Cosine of a Double Angle
Recall that cos 2q = 2cos2q - 1
Substituting in q/2 for q, you obtain
cos 2(q/2) = 2cos2(q/2) - 1
Simplifying and solving for cos (q/2)

6. Deriving Half-Angle Formulas Using Tangent of a Double Angle
Recall that tan q= sin q/cos q
Dividing the half-angle formulas for sine by the half-angle formulas for cosine you obtain *

7. Other forms of tangent of a half-angle
Rationalizing the denominator Rationalizing the numerator

8. Ex 1 Use half-angle identities to find an exact value without a calculator
sin 15
tan 195
cos (5p/12)

9. cos2x = sin2 (x/2) sin2x = cos2 (x/2)
Ex 2 Use the half-angle identities to find all solutions in the interval [0,2p)
cos2x = sin2 (x/2)
sin2x = cos2 (x/2)

10. Exit Ticket
Discuss the following questions with a neighbor and submit before leaving class: pg #