Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.
A) $- \frac { \sqrt { 11 } } { 6 }$
B) $\frac { \sqrt { 33 } } { 4 }$
C) $- \frac { 5 \sqrt { 29 } } { 29 }$
D) $\frac { 4 } { 5 }$

Question

Quizplus · Accepted Answer

In quadrant III, both sine and cosine are negative. Given $\tan \theta = \frac{2}{5}$, we can represent this as $\frac{opposite}{adjacent}$ in a right triangle. To find $\cos \theta$, we need the adjacent side over the hypotenuse. Using the Pythagorean theorem, the hypotenuse $h = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}$. Thus, $\cos \theta = \frac{adjacent}{hypotenuse} = \frac{-5}{\sqrt{29}}$, but to rationalize the denominator, it becomes $-\frac{5\sqrt{29}}{29}$.

Use the Fundamental Identities to Find the Value of the Trigonometric