Using a sum or difference identity, write the following as an expression involving functions of x.
-Find $\cos \theta$ if $\sin \theta = - \frac { 5 } { 13 }$ and $\theta$ is in quadrant III.
A) $- \frac { 5 } { 12 }$
B) $- \frac { 12 } { 13 }$
C) $\frac { 12 } { 5 }$
D) $- \frac { 13 } { 5 }$

Question

Quizplus · Accepted Answer

In the third quadrant, both sine and cosine are negative. Given $\sin \theta = -\frac{5}{13}$, we can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\cos \theta$. Plugging in the given sine value, we get $(- \frac{5}{13})^2 + \cos^2 \theta = 1$. Solving for $\cos^2 \theta$ gives $\cos^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}$, so $\cos \theta = \pm \frac{12}{13}$. Since $\theta$ is in the third quadrant where cosine is negative, $\cos \theta = -\frac{12}{13}$.

Using a Sum or Difference Identity, Write the Following as an Expression