10-3 Double-Angle and Half-Angle

Double-angle Formulas:

Sin2α = 2sinαcosα cos2α = cos^2α-sin^2α cos2α = 2cos^2α-1 cos2α = 1-2sin^2α

Half-angle Formulas:

Tan2α = +/- 2tanα/1-tan^2α sinα/2 = +/- sqrt1-cosα/2 cosα/2 = +/- sqrt1+cosα/2

Tanα/2 = +/- sqrt1-cosα/1+cosα tanα/2 = sinα/1+cosα , 1-cosα/sinα

Ex. If sinα = 4/5 and 0<α

Sin2α =2sinαcosα

Already have sin so replace it – sin2α = 2(4/5)cosα

To find cosα just draw the unit circle and find out which quadrant it’s in with 0<α

Draw a triangle with that and it’s a 3, 4, 5 triangle. Cosα= 3/5

Sin2α = 2(4/5)(3/5) = 24/25

Cos2α = 1-2sin^2α = 1-2(4/5)^2 = -7/25

Go back to your triangle and find out that tanα = 4/3

Tan2α = 2tanα/1-tan^2α = 2(4/3)/1-(4/3)^2 = 8/3/-7/9 = -72/21