Consider this equation sin(theta) = -4 square root of 29/29 if theta is an angle in quadrant IV what is the value of cos (theta)

Question

asked Aug 11, 2022 132k views

1 Answer

NKorotkov · Answer 1 · 2022-08-17T21:58:05+0000

Answer:

$\cos(\theta)= (√(377))/(29)$

Explanation:

Given

$\sin(\theta) = -(4)/(√(29))$ -- the correct expression

Required

$\cos(\theta)$

We know that:

$\sin^2(\theta) + \cos^2(\theta)= 1$

Make
$\cos^2(\theta)$ the subject

$\cos^2(\theta)= 1 - \sin^2(\theta)$

Substitute:
$\sin(\theta) = -(4)/(√(29))$

$\cos^2(\theta)= 1 - (-(4)/(√(29)))^2$

Evaluate all squares

$\cos^2(\theta)= 1 - ((16)/(29))$

Take LCM

$\cos^2(\theta)= (29 - 16)/(29)$

$\cos^2(\theta)= (13)/(29)$

Take square roots of both sides

$\cos(\theta)= \±\sqrt{(13)/(29)}$

cosine is positive in the 4th quadrant;

So:

$\cos(\theta)= \sqrt{(13)/(29)}$

Split

$\cos(\theta)= (√(13))/(√(29))$

Rationalize

$\cos(\theta)= (√(13))/(√(29)) * (√(29))/(√(29))$

$\cos(\theta)= (√(13*29))/(29)$

$\cos(\theta)= (√(377))/(29)$