Question

given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\(\sin 135^{\circ}\\)

Explanation:

Step1: Determine the reference angle

The angle $135^\circ$ is in the second quadrant. The reference angle for an angle $\theta$ in the second quadrant is $180^\circ - \theta$. So, the reference angle for $135^\circ$ is $180^\circ - 135^\circ = 45^\circ$.

Step2: Recall the sine value of the reference angle

We know that $\sin 45^\circ = \frac{\sqrt{2}}{2}$.

Step3: Determine the sign of the sine value

In the second quadrant, the sine function (which corresponds to the y - coordinate on the unit circle) is positive. So, $\sin 135^\circ=\sin(180^\circ - 45^\circ)=\sin 45^\circ=\frac{\sqrt{2}}{2}$ (using the identity $\sin(180^\circ - \alpha)=\sin\alpha$ for any angle $\alpha$).

Answer:

$\frac{\sqrt{2}}{2}$