Find the exact value of sin 105.

Steps to Solve

1. Express 105 degrees using known angles

We can express 105 degrees as the sum of 60 degrees and 45 degrees: $$ 105^\circ = 60^\circ + 45^\circ $$

2. Use the angle addition formula for sine

The sine addition formula states: $$ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) $$ In our case, we substitute (a = 60^\circ) and (b = 45^\circ): $$ \sin(105^\circ) = \sin(60^\circ)\cos(45^\circ) + \cos(60^\circ)\sin(45^\circ) $$

3. Substitute known sine and cosine values

From the unit circle or common angle values, we know:

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

Substituting these values in: $$ \sin(105^\circ) = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$

4. Simplify the expression

Now calculate each part: $$ \sin(105^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$ Combine the fractions: $$ \sin(105^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$