use the given information about θ to find the exact values of sin(2θ) , sin (θ/2), cos(2θ) ,cos (θ/2),tan(2θ) ,tan(θ/2) of cos(θ) = 3/5 where 0 < θ < π/2

Question

Hello i am stuck on this problem where i have to find all the values i listed with cos(3/5) and it lies in the first quadrant

Answer 1

theta is in quadrant I or IV, if cosine >0

sin(2theta) = 2sin(theta)cos(theta) = 2(4/5)(3/5) = 24/25 or -24/25

cos(2theta) = 1-sin^2(theta) = 1-2(4/5)^2 = 1-32/25 = -7/25

or = cos^2(theta) - sin^2(theta) = 9/25 - 16/25 = -7/25

sin(theta/2) = + or - sqr((1-cos(theta))/2) = + or - sqr(1/5) = + or - 1/sqr5

cos(theta/2) = + or - sqr(1+cos(theta))/2) = + or - sqr(4/5) = + or - 2/sqr5

tan(2theta) = sin(2theta)/cos(2theta) = + or -24/25/-7/25 = + or - 24/7

tan(theta/2) = sin(theta/2)/cos(theta/2) = + or - 1/2

if you just want values when theta is in quadrant I, ignore all the negative signs.

Answer 2

Use the trigonometric identities:

sin(2x) = 2 sin x cos x 

cos(2x) = cos^{^2}(x) - sin^{^2}(x) = 2 cos^{^2}(x) - 1 = 1 - 2 sin^{^2}(x)