Question

if if \\( \tan \theta = \frac { 3 } { \sqrt { 135 } } \\), what is \\( \cos \theta \\)? \\( \cos \theta = \frac { \sqrt { 135 } } {? } \\)

Explanation:

Step1: Recall the definition of tangent

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Given $\tan\theta = \frac{3}{\sqrt{135}}$, so the opposite side $= 3$ and the adjacent side $=\sqrt{135}$.

Step2: Use the Pythagorean theorem to find the hypotenuse

Let the hypotenuse be $c$. By the Pythagorean theorem $c^{2}=(\text{opposite})^{2}+(\text{adjacent})^{2}$.
Substitute the values: $c^{2}=3^{2}+(\sqrt{135})^{2}=9 + 135=144$. Then $c=\sqrt{144} = 12$.

Step3: Recall the definition of cosine

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Since the adjacent side $=\sqrt{135}$ and the hypotenuse $=12$.

Answer:

$12$