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.

Trigonometric Functions as Projections

The cosine function

Definition

Exercises

Check your answers