Question

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

Answer 1

C.

Explanation:

Answer 2

ANSWER

The correct answer is C


`\sin ( \theta) = - ( √(33) )/(7) , \tan ( \theta) = - ( √(33) )/( 4 )`



EXPLANATION

It was given that,



`\cos( \theta) = (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,


`( { (4)/(7) })^(2) + \sin ^(2) ( \theta) = 1`



`{ (16)/(49) } + \sin ^(2) ( \theta) = 1`



`\sin ^(2) ( \theta) = 1 - { (16)/(49) }`




`\sin ^(2) ( \theta) = { (33)/(49) }`



`\sin ( \theta) = \pm \sqrt{{ (33)/(49) }}`



`\sin ( \theta) = \pm ( √(33) )/(7)`


But


`\csc( \theta) < 0`


This implies that,



`\sin ( \theta) = - ( √(33) )/(7)`


`\tan ( \theta) = ( \sin( \theta) )/( \cos( \theta) )`

`\tan ( \theta) = ( - ( √(33) )/(7) )/( (4)/(7) )`



`\tan ( \theta) = - ( √(33) )/( 4 )`