Question

Find the value of tan( π + θ) if θ terminates in Quadrant III and sinθ = -5/13

Answer 1

ANSWER

$\tan(\pi + \theta)= \frac{5}{12}$

EXPLANATION

We first obtain

$\cos( \theta)$

using the Pythagorean Identity.

$\cos ^{2} ( \theta) + \sin ^{2} ( \theta) = 1$

$\implies \: \cos ^{2} ( \theta) + ( - \frac{5}{13} )^{2} = 1$

$\implies \: \cos ^{2} ( \theta) + \frac{25}{169}= 1$

$\implies \: \cos ^{2} ( \theta) = 1 - \frac{25}{169}$

$\implies \: \cos ^{2} ( \theta) = \frac{144}{169}$

$\implies \: \cos ( \theta) = \pm \: \sqrt{\frac{144}{169} }$

$\implies \: \cos ( \theta) = \pm \: \frac{12}{13}$

In the third quadrant, the cosine ratio is negative.

$\implies \: \cos ( \theta) = - \: \frac{12}{13}$

The tangent function has a period of π and $\pi + \theta$ is in the third quadrant.

This implies that:

$\tan(\pi + \theta)= \tan( \theta)$

$\tan(\pi + \theta)= \frac{ \sin( \theta) }{ \cos( \theta) }$

$\tan(\pi + \theta)= \frac{ - \frac{ 5}{13} }{ - \frac{12}{13} }$

This gives us:

$\tan(\pi + \theta)= \frac{5}{12}$