# How does $\cos(\pi ) = -1$?

I know this is a very elementary question, but how does $\cos(\pi) = -1$? I thought the cosine function required a minimum of 2 numbers, the adjacent side and hypotenuse of a triangle?

• The cosine function is a function of one variable, an angle. But, in terms of a right triangle, that one angle can be defined in terms of 2 side lengths. And, this is a lot simpler. So, this is what you learn at a lower level. – Graphth Apr 22 '12 at 2:01

When the angle is $0$ or $\pi$ we have a degenerate triangle, where the base and hypotenuse are in fact one and the same, except the hypotenuse has absolute length $1$ while the base, going backwards on the $x$-axis when $\theta=\pi$, has signed negative length, whence the $-1$.

Note that trigonometric functions can be described geometrically by ratios of signed lengths of triangle figures drawn on a circle, but these ratios are completely determined by the angle and so a trig function is purely a function of that angle.

• Thank you for the clear and concise explanation. – subtlearray Apr 22 '12 at 2:31

Any point on the unit circle centered at $(0,0)$ can be parametrised by $\theta$ such that $0 \le \theta \lt 2 \pi$, such that $$x=\cos \theta\\y=\sin \theta$$

Geometrically, $\theta$ is the angle made by the vector joining $(\cos \theta, \sin \theta)$ and $(0,0)$ with the $x-$axis.

Can you cook these facts to see your result?

Hint You may want to use the picture below. (Courtesy: Wikipedia)

$\hskip{2 in}$ • Seeing the unit circle helped put this into better perspective. I see now why my teacher prefers it over using triangles. Thank you for your reply. – subtlearray Apr 22 '12 at 2:21
• Glad to have been of some help here. :) @SubtleArray – user21436 Apr 22 '12 at 2:22

Cosine is a function where you put one number in (input) and you get one number out (output). The question is: how to I know how to find the number that comes out when I know what goes in. For cosine of a number (an angle) we can "draw" a triangle like below where the angle at $A$ is the input. Call the input $A$. Then we have that $\cos(A) = \frac{h}{b}$. So in a certain sense, the output is given by the two numbers (hypotenuse) and (adjacent), but we just have one angle - one input. I like to think of it like this: If you look in the picture above, it is clear that $\cos \theta$ is a positive number. If $\pi/2 < \theta < 3\pi/2$ $\cos \theta$ is negative, because its length is in fact negative as the line representing $\cos$ is left of the y axis.

If you are familiar with radians, $\pi$ is $180$ degrees. This means, if I were to draw the image above with $\theta=\pi$, the line would be horizontal, one point connected to the leftmost point on the circle, the other attached to the radius. $\sin \theta$ is zero, as the height (y) has a length of zero. $\cos \theta$ however is -1, because it has a length of negative 1.

This is sort of hard to explain without an animation or another drawing, but it is a simple concept once understood.