# How do you find the exact value of arccos(-1/sqrt(2))?

Oct 14, 2016

$\frac{3 \pi}{4}$

#### Explanation:

$\arccos \left(- \frac{1}{\sqrt{2}}\right)$

First, it would be helpful to rationalize $- \frac{1}{\sqrt{2}}$ because unit circle values are usually rationalized.

$- \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = - \frac{\sqrt{2}}{2}$

Arccos is asking for the ANGLE with a cosine of the given value.

The range of arccos is between zero and $\pi$. So if you are finding an arccos of a positive value, the answer is between zero and $\frac{\pi}{2}$. If you are finding the arccos of a negative value, the answer is between $\frac{\pi}{2}$ and $\pi$.

According to the unit circle, the angle in the second quadrant (between $\frac{\pi}{2}$ and $\pi$) with a cosine of $- \frac{\sqrt{2}}{2}$ is $\frac{3 \pi}{4}$.

Jan 16, 2017

$\frac{3 \pi}{4} , \frac{5 \pi}{4}$

#### Explanation:

cos x = - 1/(sqrt2) = - sqrt2/2
On the trig unit circle, there are 2 arcs that have the same cos value:
$x = \frac{3 \pi}{4}$ and $x = \frac{5 \pi}{4}$
Answers for $\left(0 , 2 \pi\right)$:
$\frac{3 \pi}{4} , \frac{5 \pi}{4}$
Check with calculator:
$\cos \left(\frac{3 \pi}{4}\right) = \cos {135}^{\circ} = - 0.707$
$\cos \left(\frac{5 \pi}{4}\right) = \cos {225}^{\circ} = - 0.707$.

Feb 26, 2017

$\frac{3 \pi}{4}$

#### Explanation:

color(blue)(arccos(-1/(sqrt2))

First we should understand what the question is about.

It means that, we need to find an angle, when it is inside a cosine function gives $- \frac{1}{\sqrt{2}}$

Let's rationalize it

$\rightarrow - \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$\Rightarrow - \frac{\sqrt{2}}{2}$

Now let's find out the angle using the unit circle

The angle is color(green)((3pi)/(4)