Let the angle θ satisfy sin 2θ=4/5 and (3π/4)<θ<π . Find the value of: P=sin θ - cos θ

Question

Sidenote: Please explain your answer step-by-step, thank you! ^^

Answer 1

Given that sin(2θ) = - 4/5, the double angle formula for sine is sin(2θ) = 2 sin(θ) cos(θ),

and the Pythagorean identity for sine and cosine is sin²(θ) + cos²(θ) = 1, we have the following:

P² = [ sin(θ) - cos(θ) ]² = sin²(θ) - 2 sin(θ) cos(θ) + cos²(θ) = [ sin²(θ) + cos²(θ) ] - [ 2 sin(θ) cos(θ) ],

so substituting the given information in, we get the following:

P² = 1 - (- 4/5) = 5/5 + 4/5 = 9/5, so P = +- √(P²) = +- √(9/5) = +- 3 / √(5),

but for 3π/4 < θ < π, we have that sin(θ) is > 0 and cos(θ) < 0, so P = sin(θ) - cos(θ)

is definitely > 0, so we finally have that P = + 3 / √(5).

Answer 2