Express cos 150° in surd form.

Here are the solutions to the problems:

1. Find $\sin 135^\circ$

Step 1: Express $\sin 135^\circ$ using the sum of angles formula. We can write $135^\circ$ as $90^\circ + 45^\circ$. The sum identity for sine is $\sin(a+b) = \sin a \cos b + \cos a \sin b$. $\sin 135^\circ = \sin(90^\circ + 45^\circ)$ Step 2: Apply the sum identity. $\sin(90^\circ + 45^\circ) = \sin 90^\circ \cos 45^\circ + \cos 90^\circ \sin 45^\circ$ Step 3: Substitute the known values for the trigonometric functions. We know that $\sin 90^\circ = 1$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 90^\circ = 0$, and $\sin 45^\circ = \frac{\sqrt{2}}{2}$. $= (1)\left(\frac{\sqrt{2}}{2}\right) + (0)\left(\frac{\sqrt{2}}{2}\right)$ Step 4: Simplify the expression. $= \frac{\sqrt{2}}{2} + 0$ $= \frac{\sqrt{2}}{2}$ The value of $\sin 135^\circ$ is $\boxed{\frac{\sqrt{2}}{2}}$.

2. Express $\cos 150^\circ$ in surd form.

Step 1: Express $\cos 150^\circ$ using the sum of angles formula. We can write $150^\circ$ as $90^\circ + 60^\circ$. The sum identity for cosine is $\cos(x+y) = \cos x \cos y - \sin x \sin y$. $\cos 150^\circ = \cos(90^\circ + 60^\circ)$ Step 2: Apply the sum identity. $\cos(90^\circ + 60^\circ) = \cos 90^\circ \cos 60^\circ - \sin 90^\circ \sin 60^\circ$ Step 3: Substitute the known values for the trigonometric functions. We know that $\cos 90^\circ = 0$, $\cos 60^\circ = \frac{1}{2}$, $\sin 90^\circ = 1$, and $\sin 60^\circ = \frac{\sqrt{3}}{2}$. $= (0)\left(\frac{1}{2}\right) - (1)\left(\frac{\sqrt{3}}{2}\right)$ Step 4: Simplify the expression. $= 0 - \frac{\sqrt{3}}{2}$ $= -\frac{\sqrt{3}}{2}$ The value of $\cos 150^\circ$ in surd form is $\boxed{-\frac{\sqrt{3}}{2}}$.