# How do you tell whether the value of cos [5(pi)/4] is positive, negative, zero or undefined?

Oct 16, 2015

It's negative.

#### Explanation:

First of all, the cosine is defined all over the real numbers set, so $\cos \left(x\right)$ is defined for every $x$. Since the cosine is the projection on the $x$-axis of the points on the unit circle, the cosine is positive if the point on the unit circle is on the first or fourth quadrant, and is negative if the point is in the second or third quadrant. Cosine's roots are integer multiples of $\frac{\pi}{2}$ and $\frac{3 \pi}{2}$.

So, $\frac{5 \pi}{4}$ is $\pi + \frac{\pi}{4}$, so $\frac{5 \pi}{4}$ is $\frac{\pi}{4}$ radians more than a straight angle.

Visually, this means that $\frac{5 \pi}{4}$ identifies the midpoint between the "west pole" and the "south pole", and so it's in the third quadrant.

Thus, its cosine is negative.