Question

An angle θ on the unit circle has a sine of 14. What is the value of cosθ?

Answer 1

Given:

An angle θ on the unit circle has a sine of $\dfrac{1}{4}$.

To find:

The value of cosθ.

Solution:

We have,

$\sin \theta=\dfrac{1}{4}$

Since sinθ is positive, therefore, θ lies in first or second quadrant.

We know that,

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

$(\dfrac{1}{4})^2+\cos^2 \theta =1$

$\dfrac{1}{16}+\cos^2 \theta =1$

$\cos^2 \theta =1-\dfrac{1}{16}$

Taking square root on both sides.

$\cos \theta =\pm \sqrt{\dfrac{16-1}{16}}$

$\cos \theta =\pm \dfrac{\sqrt{15}}{4}$

Therefore, the value of cosθ is either $\dfrac{\sqrt{15}}{4}$ or $-\dfrac{\sqrt{15}}{4}$.