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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.

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
*y*coordinate of point P.

- 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.

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