Find the exact value of the expression using the provided information.
-Find $\sin \theta$ if $\tan \theta = - \frac { 12 } { 5 }$ and $\cos \theta 0$.
A) $- \frac { 5 } { 13 }$
B) $\frac { 5 } { 13 }$
C) $- \frac { 13 } { 12 }$
D) $- \frac { 12 } { 13 }$

Question

Find the exact value of the expression using the provided information.
-Find $\sin \theta$ if $\tan \theta = - \frac { 12 } { 5 }$ and $\cos \theta > 0$.
A) $- \frac { 5 } { 13 }$
B) $\frac { 5 } { 13 }$
C) $- \frac { 13 } { 12 }$
D) $- \frac { 12 } { 13 }$

Quizplus · Accepted Answer

Given $\tan \theta = -\frac{12}{5}$, we can represent this as opposite/adjacent in a right triangle, implying the sides are 12 (opposite) and 5 (adjacent). Since $\cos \theta > 0$, $\theta$ is in the fourth quadrant where cosine is positive and sine is negative. The hypotenuse can be found using the Pythagorean theorem: $\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$. Thus, $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-12}{13}$, because sine is negative in the fourth quadrant.

Find the Exact Value of the Expression Using the Provided sin⁡θ\sin \thetasinθ

Find the Exact Value of the Expression Using the Provided $\sin \theta$