Question

Please help me

If 180°<α<270°, cos⁡ α=−8/17, what is sin -α?

Answer 1

15/17

Explanation:

opposite² = 17² - 8²

opposite² = 225

opposite = 15

sin a = - 15/17

(sin is -ve in the third quadrant)

sin(-X) = -sinX

sin(-a) = -sin(a)

= -(-15/17)

= 15/17

Answer 2

Starting from the fundamental trigonometric equation, we have

`\cos^2(\alpha)+\sin^2(\alpha)=1 \iff \sin(\alpha)=\pm√(1-\cos^2(\alpha))`

Since
`180<\alpha<270`, we know that the angle lies in the third quadrant, where both sine and cosine are negative. So, in this specific case, we have

`\sin(\alpha)=-√(1-\cos^2(\alpha))`

Plugging the numbers, we have

`\sin(\alpha)=-\sqrt{1-(64)/(289)}=-\sqrt{(225)/(289)}=-(15)/(17)`

Now, just recall that

`\sin(-\alpha)=-\sin(\alpha)`

to deduce

`\sin(-\alpha)=-\sin(\alpha)=-\left(-(15)/(17)\right)=(15)/(17)`