Question

If tan theta equals 15/8 in quadrant 3 and you need to find the sin, cos, and tan of double angle, in what quadrant would the double angle reside.

Answer 1

Answer: sin2θ = 240/289

cos2θ = -161/289

tan2θ = -240/161

2θ is in quadrant 2

Step-by-step explanation: tanθ = 15/8

sinθ/cosθ = 15/8

sinθ = 15cosθ/8

sin²θ + cos²θ = 1

(15cosθ/8)² + cos²θ = 1

225cos²θ/64 + cos²θ = 1

225cos²θ + 64cos²θ = 64

289cos²θ = 64

cos²θ = 64/289

cosθ = ±√64/289

cosθ = ±8/17

As tanθ in quad 3 ⇒ tanθ is +, cosθ is - and sinθ is -

So, cosθ = -8/17

as sin θ = 15cosθ/8 = -15/17

For sin2θ = 2sinθcosθ = 2.(-15/17).(-8/17) = 240/289

For cos2θ = cos²θ - sin²θ = (-8/17)² - (-15/17)² = -161/289

For tan2θ = sin2θ/cos2θ = (240/289) / (-161/289) = -240/161

As sin2θ + and cos2θ - ⇒ 2θ is in quadrant 2