2tanu tan2u 1 tan - utep mathematicssin2 1 cos2 2 u u cos2 1 cos2 2 u u tan2 1 cos2 1 cos2 u u...
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5.5 Multiple-Angle and Product-to-Sum Formulas
Double-Angle Formulas
sin2 2sin cosu u u 2
2 tantan 2
1 tan
uu
u
2 2
2
2
cos2 cos sin
2cos 1
1 2sin
u u u
u
u
Example: Solve 2cos sin2 0x x
Example: If5
cos13
, with theta in the fourth quadrant, find sin 2 ,cos2 , and tan 2 .
Power-Reducing Formulas
2 1 cos2sin
2
uu
2 1 cos2
cos2
uu
2 1 cos2
tan1 cos2
uu
u
Half-Angle Formulas
1 cos
sin2 2
u u
1 coscos2 2
u u
1 cos sintan2 sin 1 cos
u u u
u u
The signs of the sine and cosine depend on the quadrant that u/2 lies in.
Example: Find the exact value ofsin105 .
Example: Find all solutions of 2 22 sin 2cos2
xx in the interval [0,2 ) .
Product-to-Sum Formulas
1
sin sin cos cos2
u v u v u v 1
cos cos cos cos2
u v u v u v
1
sin cos sin sin2
u v u v u v 1
cos sin sin sin2
u v u v u v
Sum-to-Product Formulas
sin sin 2sin cos2 2
u v u vu v
sin sin 2cos sin
2 2
u v u vu v
cos cos 2cos cos2 2
u v u vu v
cos cos 2sin sin
2 2
u v u vu v
Example: Find the exact value ofcos195 cos105 .