# How do you convert (2, pi/4) into rectangular coordinates?

Jan 18, 2016

$x = 2 \cos \left(\frac{\pi}{4}\right) = 2 \cdot 0.707 = 1.414$

$y = 2 \sin \left(\frac{\pi}{4}\right) = 2 \cdot 0.707 = 1.414$

Rectangular coordinates are (1.4,1.4)

#### Explanation:

These coordinates describe a line 2 units long, starting at the origin, $\left(0 , 0\right)$, at an angle of $\frac{\pi}{4}$ radians anticlockwise (counterclockwise) from the positive axis.

Some find it easier to work in radians, some in degrees. $\frac{\pi}{4}$ is ${45}^{o}$. I'll keep working in radians, since that is how the question is set up.

For rectangular coordinates we need to find the distance of the projection along the x-axis for the first point and along the y-axis for the second.

Draw a diagram. It's crucial.

Now use the definition of trigonometry. The two points are as follows:

$x = 2 \cos \left(\frac{\pi}{4}\right) = 2 \cdot 0.707 = 1.414$

$y = 2 \sin \left(\frac{\pi}{4}\right) = 2 \cdot 0.707 = 1.414$

Jan 18, 2016

$\left(\sqrt{2} , \sqrt{2}\right)$

#### Explanation:

Using the formulae that links Polar and Cartesian coordinates .

• x = rcostheta

• y = rsintheta

here r = 2 , $\theta = \frac{\pi}{4}$

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

and y = $2 \sin \left(\frac{\pi}{4}\right) = 2. \frac{1}{\sqrt{2}} = \sqrt{2}$