cos ( 2θ ) = 0.66 The angle 2θ is in quadrant IV cos ( θ ) = ?

Question

Using a double-angle or half-angle formula to find 

Answer 1

cos(theta) = cos(2theta/2) = -sqr((1+cos2theta)/2) = -sqr(1.66/2) = -sqr(0.83) = -0.911

use the half angle formula

or find the angle whose cosine is 0.66 in quadrant IV = -48.7 degrees = 360-48.7 = 311.3

divide by 2: 155.65, take the cos(155.65) = -0.911

Answer 2

Use a half-angle formula to find Cos(θ):



https://precalculusfinal.weebly.com/18-half-angle-identities.html

(Unfortunately, this editor does not allow you to post images so I can’t show the formulas directly.)

But these formulas involve square roots. So you need to say if it’s the positive or negative root.

If 2θ is in Quadrant IV, then 270° ≤ 2θ ≤ 360°. So 135° ≤ θ ≤ 180°, which is the 2nd quadrant. In that quadrant, cos is always negative.

Robert C.