ANSWER
[tex] { \sin ( \theta) } = -\frac{3}{5} [/tex]
EXPLANATION
The cosine function is an even function.
It has the following property.
[tex] \cos( - \theta) = \cos( \theta) [/tex]
This implied that if
[tex] \cos( - \theta) = \frac{4}{5} [/tex]
Then,
[tex] \cos( \theta) = \frac{4}{5} [/tex]
Using the Pythagorean identity,
[tex] { \cos ^{2} ( \theta) } + { \sin ^{2} ( \theta) } = 1[/tex]
We substitute the value of
[tex] \cos( \theta) = \frac{4}{5} [/tex]
into the equation to get,
[tex] ( { \frac{4}{5} })^{2} + { \sin ^{2} ( \theta) } = 1[/tex]
[tex] \frac{16}{25}+ { \sin ^{2} ( \theta) } = 1[/tex]
[tex] { \sin ^{2} ( \theta) } = 1 - \frac{16}{25} [/tex]
[tex] { \sin ^{2} ( \theta) } = \frac{9}{25} [/tex]
We take square root of both sides to get,
[tex] { \sin ( \theta) } = \pm \: \sqrt{ \frac{9}{25} } [/tex]
[tex] { \sin ( \theta) } = \pm \frac{3}{5} [/tex]
It was given that,
[tex] \tan( \theta) \: > \: 0[/tex]
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,
[tex] { \sin ( \theta) } = \frac{3}{5} [/tex].
But the sine function is an odd function.
This means that,
[tex] \sin( - \theta) =-\sin( \theta)[/tex]
Therefore,
[tex] { \sin ( -\theta) } =- \frac{3}{5} [/tex].
