Question

Given cos theta=4/7 and csc theta<0, find sin theta and tan theta.

Answer 1

ANSWER

The correct answer is C

$\sin ( \theta) = - \frac{ \sqrt{33} }{7} , \tan ( \theta) = - \frac{ \sqrt{33} }{ 4 }$



EXPLANATION

It was given that,


$\cos( \theta) = \frac{4}{7}$


and

$\csc( \theta) < 0$

This means that,


$\theta$
is in the fourth quadrant.


We use the identity,

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


This implies that,

$( { \frac{4}{7} })^{2} + \sin ^{2} ( \theta) = 1$


${ \frac{16}{49} } + \sin ^{2} ( \theta) = 1$


$\sin ^{2} ( \theta) = 1 - { \frac{16}{49} }$



$\sin ^{2} ( \theta) = { \frac{33}{49} }$


$\sin ( \theta) = \pm \sqrt{{ \frac{33}{49} }}$


$\sin ( \theta) = \pm \frac{ \sqrt{33} }{7}$


But

$\csc( \theta) < 0$


This implies that,


$\sin ( \theta) = - \frac{ \sqrt{33} }{7}$

$\tan ( \theta) = \frac{ \sin( \theta) }{ \cos( \theta) }$
$\tan ( \theta) = \frac{ - \frac{ \sqrt{33} }{7} }{ \frac{4}{7} }$


$\tan ( \theta) = - \frac{ \sqrt{33} }{ 4 }$