### Instruction: Compute $\sin 135^\circ$.### Response: To find the sine of any angle, I need to recall the definition of the sine function in terms of the unit circle.
The unit circle is a circle with radius 1 centered at the origin, and the sine of any angle is the y-coordinate of the point on the circle that forms that angle with the positive x-axis.
So, to find $\sin 135^\circ$, I need to locate the point on the unit circle that is 135 degrees counterclockwise from the positive x-axis.
I can use the fact that 135 degrees is the same as 90 degrees plus 45 degrees, and I can use the angle sum formula for sine: $\sin(A + B) = \sin A \cos B + \cos A \sin B$.
Plugging in A = 90 and B = 45, I get $\sin 135^\circ = \sin 90^\circ \cos 45^\circ + \cos 90^\circ \sin 45^\circ$.
Now, I need to remember the values of sine and cosine for 90 and 45 degrees.
I know that $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$, and $\cos 45^\circ = \frac{\sqrt{2}}{2}$.
Substituting these values into the formula, I get $\sin 135^\circ = 1 \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$.