Question

Use 90°<θ<180° and sin θ=24/25 ⁡to answer the following questions. What is cos⁡ θ?

Answer 1

Answer:

-7/25

Step-by-step explanation:

$\theta$ is in quadrant two given that $\theta$ is between 90 degrees and 180 degrees.

This means cosine value there is negative and sine value is positive.

Let's use the Pythagorean Identity: $\sin^2(\theta)+\cos^2(\theta)=1$.

$(\frac{24}{25})^2+\cos^2(\theta)=1$

$\frac{576}{625}+\cos^2(\theta)=1$

Subtract 576/625 on both sides:

$\cos^2(\theta)=1-\frac{576}{625}$

$\cos^2(\theta)=\frac{625-576}{625}$

$\cos^2(\theta)=\frac{49}{625}$

Take the square root of both sides:

$\cos(\theta)=\pm \frac{7}{25}$

So recall that the cosine value here is negative due to the quadrant we are in.

$\cos(\theta)=-\frac{7}{25}$

Check:

$(\frac{24}{25})^2+(-\frac{7}{25})^2$

$\frac{576+49}{625}$

$\frac{625}{625}$

$1$

So we got the desired result since the right hand side of our Pythagorean Identity is 1.