Trigonometric Functions as Projections

The tangent function

The third main trigonometric function is the tangent. You probably already know one definition of the tangent of an angle as being OPPOSITE/ADJACENT in a right triangle. Let's see the projection definition of the tangent.

Definition

Refer to the applet below.

Consider a unit circle with a line T drawn tangent to the circle, touching at (1,0).

Let L be a line through the origin drawn at an angle \(\theta\) anti-clockwise from the positive x-axis.

Let P be the point defined by the intersection of T and L.

The tangent of angle \(\theta\) is defined as the value of ycoordinate of point P.

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

Exercises

Explain what happens to \(\tan(\theta)\) as \(\theta\) approaches 90° or -90°.
What is the range of the tangent function? (That is, what are the minimum and maximum values \(\tan(\theta)\) can take for any \(\theta\)?)
The x- and y- axes divide the plane into four quadrants. In which of these quadrants is \(\tan(\theta)\) 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 tangent function

Definition

Exercises

Check your answers