# What are the components of the vector between the origin and the polar coordinate (1, (5pi)/4)?

Jan 15, 2016

The $x$ component is: $\cos \left(\frac{5 \pi}{4}\right)$
The $y$ component is: $\sin \left(\frac{5 \pi}{4}\right)$

#### Explanation:

Remembering our trigonometry, the vertical component of a vector is given by
$r \cdot \sin \left(\theta\right)$ where $r$ is the length of the line,
and the horizontal component by
$r \cdot \cos \left(\theta\right)$

in the polar coordinate $\left(1 , \frac{5 \pi}{4}\right)$, $r$ is 1, and the angle $\theta = \frac{5 \pi}{4}$.

Hence:
The $x$ component is: $\cos \left(\frac{5 \pi}{4}\right)$
The $y$ component is: $\sin \left(\frac{5 \pi}{4}\right)$

In this case, $\frac{5 \pi}{4}$ is midway in the lower left quadrant, or ${135}^{\circ}$, so both are equal to $- \frac{1}{\sqrt{2}}$