exact value of sin 165

Steps to Solve

1. Identify Related Angles

We can express $165^\circ$ as the sum of two known angles: $165^\circ = 120^\circ + 45^\circ$.

2. Use Sine Addition Formula

We apply the sine addition formula:

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

In this case, let $a = 120^\circ$ and $b = 45^\circ$.

3. Evaluate Sine and Cosine for Known Angles

Now we need the sine and cosine values:

• $ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} $
• $ \cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2} $
• $ \sin(45^\circ) = \frac{\sqrt{2}}{2} $
• $ \cos(45^\circ) = \frac{\sqrt{2}}{2} $

4. Substitute Values into the Formula

Substituting the values into the sine addition formula:

$$ \sin(165^\circ) = \sin(120^\circ) \cos(45^\circ) + \cos(120^\circ) \sin(45^\circ $$

This becomes:

$$ \sin(165^\circ) = \left( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \right) + \left(-\frac{1}{2} \cdot \frac{\sqrt{2}}{2} \right) $$

5. Simplify the Expression

Now simplify the expression:

$$ \sin(165^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} $$

Combine the terms:

$$ \sin(165^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4} $$