Question

What is the exact value of sin (165 degrees)

Answer 1

Explanation:

(√6 - √2)/4

Answer 2

The exact value of
`$\sin(165^\circ)$` is
`$\frac{\sqrt{2-√(3)}}{4}$`, which is option C.

The exact value of
`$\sin(165^\circ)$` is
`$\frac{\sqrt{2-√(3)}}{4}$`. This corresponds to option C.

To find this value, we can use the angle addition formula for sine:

`$\sin(a+b) = \sin a \cos b + \cos a \sin b$`

In this case, we can rewrite
`$165^\circ$` as the sum of
`$60^\circ$` and
`$105^\circ$`. So, we have:

`$\sin(165^\circ) = \sin(60^\circ + 105^\circ)$`

Now, let's use the values of sine and cosine for
`$60^\circ$` and
`$105^\circ$`:

`$\sin(165^\circ) = \sin(60^\circ)\cos(105^\circ) + \cos(60^\circ)\sin(105^\circ)$`

`$\sin(165^\circ) = (√(3))/(2)\cdot(√(2)+√(6))/(4) + (1)/(2)\cdot(√(2)-√(6))/(4)$`

Simplifying this expression, we get:

`$\sin(165^\circ) = \frac{\sqrt{2+√(3)}}{4}$`

Therefore, option C is the correct answer.