What is the exact value of sin 75 degrees?

Steps to Solve

1. Use the sine angle addition formula

The sine of 75 degrees can be expressed as the sum of two angles:

$$ 75^\circ = 45^\circ + 30^\circ $$

Using the sine angle addition formula, we have:

$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) $$

The formula is:

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

2. Substituting the values

Now substitute $a = 45^\circ$ and $b = 30^\circ$ into the formula:

$$ \sin(75^\circ) = \sin(45^\circ) \cos(30^\circ) + \cos(45^\circ) \sin(30^\circ $$

We know the values of the trigonometric functions:

• $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
• $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
• $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
• $\sin(30^\circ) = \frac{1}{2}$

Substituting these values gives:

$$ \sin(75^\circ) = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) $$

3. Simplifying the expression

Now simplify the expression:

$$ \sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$

Combine the terms:

$$ \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$