Question

What is the sine value of 7 pi over 6? negative square root 3 over 2 square root 3 over 2 negative 1 over 2 1 over 2?

Answer 1

First thing to do is to change the radians to degrees so it's easier to determine our angle and where it lies in the coordinate plane.
`(7 \pi )/(6) * (180)/( \pi )=210`. If we sweep out a 210 degree angle, we end up in the third quadrant, with a 30 degree angle. In this quadrant, x and y are both negative, but the hypotenuse, no matter where it is, will never ever be negative. So the side across from the 30 degree reference angle is -1, and the hypotenuse is 2, so the sine of this angle, opposite over hypotenuse, is -1/2

Answer 2

-1/2.

Explanation:

Let's decompose
`(7\pi)/(6)`:

`(7\pi)/(6)=\pi+(\pi)/(6)`

So,

`sin((7\pi)/(6))=sin(\pi+(\pi)/(6))`

`=sin(\pi)cos((\pi)/(6))+sin((\pi)/(6))cos(\pi)`

`=(1)/(2)(-1)=(-1)/(2).`