# How do you find the value of cos(pi/4)?

Mar 7, 2018

You would look on the unit circle.

#### Explanation:

$\cos \left(\frac{\pi}{4}\right)$= $\left(\frac{1}{\sqrt{2}}\right) = \frac{\sqrt{2}}{2}$

!unit circle

Mar 7, 2018

$\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, refer to the explanation below for how to find the exact value without a calculator.

#### Explanation:

It is possible to find the exact value of $\cos \left(\frac{\pi}{4}\right)$ by constructing a right triangle with one angle set to $\frac{\pi}{4}$ radians.

First, let's convert radians into degrees
$\frac{\pi}{4} \text{ rad"=pi/4 " rad" * 180^"o"/(pi)* "rad"^-1=45^"o}$

Now let's draw a right triangle with one of the acute angles set to $45$ degrees. Remeber that these two angles would be supplementary, meaning that their sum would be ${90}^{\text{o}}$. As a result, the other angle in this triangle would also be ${45}^{\text{o}}$, making an isosceles right triangle.

By setting the length of one of the sides adjacent to the right angle to $1$ and applying the Pythagorean theorem, you'll find the length of the hypotenuse $\sqrt{{1}^{2} + {1}^{2}} = \sqrt{2}$.

Thus $\cos \left(\frac{\pi}{4}\right) = \cos \left({45}^{\text{o")=("adj.")/("hyp.}}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.