θ and a are acute angles in standard position. sin θ = 3/5 and cos a = 12/13. Determine the exact value of cos(a+θ).

Question

Also please explain why the direct angle measurements are not needed to find the compound trigonometric ratios.

Answer 1

cos(a+t) = cos^2(a) - sin^2(t) = (12/13)^2 - (3/5)^2 = 144/169- 9/25 = about 1/2

a= about 37

t = about 23

a+t= about 60

cos(60) = 1/2

cos(A+B) = cosAcosB - sinAsinB

if A=B then

cos2A = cos^2A - sin^2A

for example

cos(60) = cos^2(30)-sin^2(30

1/2 = [(sqr3)/2] - [1/2]^2 = 3/4-1/4 = 1/2