Question

Let cos(−θ)=4/5 and tanθ> 0. What is the value of sin(−θ)?

Answer 1

`-(3)/(5)`

Explanation:

Answer 2

ANSWER

`{ \sin ( \theta) } = -(3)/(5)`

EXPLANATION

The cosine function is an even function.

It has the following property.


`\cos( - \theta) = \cos( \theta)`

This implied that if


`\cos( - \theta) = (4)/(5)`

Then,


`\cos( \theta) = (4)/(5)`

Using the Pythagorean identity,


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

We substitute the value of

`\cos( \theta) = (4)/(5)`

into the equation to get,


`( { (4)/(5) })^(2) + { \sin ^(2) ( \theta) } = 1`


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


`{ \sin ^(2) ( \theta) } = 1 - (16)/(25)`


`{ \sin ^(2) ( \theta) } = (9)/(25)`

We take square root of both sides to get,


`{ \sin ( \theta) } = \pm \: \sqrt{ (9)/(25) }`


`{ \sin ( \theta) } = \pm (3)/(5)`

It was given that,


`\tan( \theta) \: > \: 0`

This implies that the angle is in the first quadrant. That is the only quadrant where both the cosine and the tangent ratios are positive.

Hence,


`{ \sin ( \theta) } = (3)/(5)`.

But the sine function is an odd function.

This means that,


`\sin( - \theta) =-\sin( \theta)`

Therefore,


`{ \sin ( -\theta) } =- (3)/(5)`.