# Trigonometric Functions as Projections

## The cosine function

As with the sine, you probably already know one definition of the
cosine of an angle as being ADJACENT/HYPOTENUSE in a right
triangle. Again, this definition is fine, as far as it goes, but
suffers from the same limitations as the triangle definition for the
sine. The projection definition of the cosine is very similar to that
of the sine.

### Definition

Refer to the interactive below.

Let *P* be a point defined by the intersection of a ray
drawn at an angle \(\theta\) anti-clockwise from the positive
*x-*axis with the unit circle.

Project a line parallel to the
*y-*axis and passing through *P*.

The cosine of angle
\(\theta\) is defined as the value of
*x* where this line passes through the *x-*axis.

Use the interactive to explore the value of \(\cos(\theta)\) for different
values of \(\theta\).

## Exercises

- What is the
*range* of the cosine function? (That is, what are
the minimum and maximum values \(\cos(\theta)\) can take for any
\(\theta\)?)
- The
*x-* and *y-* axes divide the plane into four
quadrants. In which of these quadrants is \(\cos(\theta)\) positive,
and in which is it negative? Sketch a diagram with the
*x-* and *y-* axes shown, and put a '+' or '-' in each
quadrant.