cos 15 exact value

Steps to Solve

1. Use the Cosine Difference Formula

To find $\cos(15^\circ)$, we can express it using a known angle: $$ \cos(15^\circ) = \cos(45^\circ - 30^\circ) $$ We will apply the cosine difference identity: $$ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) $$

2. Substitute Known Values

Next, we substitute $a = 45^\circ$ and $b = 30^\circ$. So, $$ \cos(15^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ $$

3. Calculate Each Trigonometric Function

Now we need the values of the trigonometric functions at these angles:

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

Plugging these values into our equation gives: $$ \cos(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$

4. Combine the Terms

Now we simplify the expression: $$ \cos(15^\circ) = \frac{\sqrt{2}\sqrt{3}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} $$