cos 15 exact value

Understand the Problem

The question is asking for the exact value of cos(15 degrees). This involves using trigonometric identities or formulas to find the value without using a calculator.

Answer

$$ \frac{\sqrt{6} + \sqrt{2}}{4} $$

Steps to Solve

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

Substitute Known Values

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

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

Combine the Terms

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

The exact value of $\cos(15^\circ)$ is given by $$ \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$

More Information

The cosine values for angles like 15 degrees can be derived using trigonometric identities such as the cosine difference formula. This is a useful method when calculators aren't available. Knowing these angles and their cosine values can also aid in solving more complex problems.

Tips

A common mistake is to confuse the angles when applying the cosine difference formula. It's important to correctly identify the angles being used.
Forgetting to apply the signs correctly when combining terms can lead to incorrect results. Pay attention to addition and ensure all terms are correctly calculated.

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