To find the exact value of $\cos(15^\circ)$ using a half-angle formula, we can start by identifying the appropriate formula and angle.

Part 1: Determine the Half-Angle Formula

The half-angle formula for cosine is:

$\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}}$

To find $\cos(15^\circ)$, we can express $15^\circ$ as half of $30^\circ$:

$15^\circ = \frac{30^\circ}{2}$

Thus, we can use the half-angle formula with $\theta = 30^\circ$:

$\cos(15^\circ) = \sqrt{\frac{1 + \cos(30^\circ)}{2}}$

Part 2: Calculate $\cos(15^\circ)$

Next, we find $\cos(30^\circ)$:

$\cos(30^\circ) = \frac{\sqrt{3}}{2}$

Now substitute this value back into the half-angle formula:

$\cos(15^\circ) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}$

Simplifying the expression inside the square root:

1. Combine the terms in the numerator:

$1 + \frac{\sqrt{3}}{2} = \frac{2}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$

2. Now substitute back:

$\cos(15^\circ) = \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}$

Thus, the exact value is:

$\cos(15^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2}$

Summary

• Half-Angle Formula: $\cos(15^\circ) = \sqrt{\frac{1 + \cos(30^\circ)}{2}}$
• Exact Value: $\cos(15^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2}$

Questions for Further Exploration

1. What are the half-angle formulas for sine and tangent?
2. How can you derive the value of $\cos(45^\circ)$ using a similar method?
3. What other angles can be expressed using half-angle formulas?
4. Can you explain why the cosine of an angle in the first quadrant is positive?
5. How does the cosine of complementary angles relate to the sine of those angles?

Tip

When using half-angle formulas, ensure you know the cosine values of common angles (like $30^\circ$, $45^\circ$, and $60^\circ$) to simplify your calculations effectively.