Double Angle(Sine/Cosine)

Question

Assume that θ is a positive acute angle.

Given: cosθ=35/37

find: sin 2θ

Answer 1

sinx = sqr(37^2-35^2)/37

=12/37

sin2x = 2sinxcosx

= 2(12)/37][35/37

Sine of Two Theta =840/1369= .614

check the answer, find the angle x

x = cos^-1(35/37)=18.925

2x=37.85

sin2x=sin38.85 = .614

Answer 2

Hello Justin,

By definition: sin 2θ = 2 sinθ cosθ

You know cosθ, so to find sinθ, you can use the Pythagorean identity sin²θ + cos²θ = 1 to find sinθ

sin²θ = 1 - cos²θ = 1 - (35/37)² = 1 - (1225/1369) = 144/1369 so sin θ = ± square root of (144/1369)

θ is a positive acute angle (aka it’s in Quadrant 1, between 0 and 90 degrees), so sin θ will equal the POSITIVE square root of 144/1369 = 12/37

now plug it into the formula I mentioned for sin 2θ:

sin 2θ = 2 (35/37)(12/37) = 2 (420/1369) = 840/1369