Tangent Function

The tangent function is a periodic function which is very important in trigonometry.

The simplest way to understand the tangent function is to use the unit circle. For a given angle measure

θ

draw a unit circle on the coordinate plane and draw the angle centered at the origin, with one side as the positive

x

-axis. The

x

-coordinate of the point where the other side of the angle intersects the circle is

\cos (θ)

and the

y

-coordinate is

\sin (θ)

There are a few sine and cosine values that should be memorized, based on $30 ° - 60 ° - 90 °$ triangles and $45 ° - 45 ° - 90 °$ triangles. Based on these, you can work out the related values for tangent.

$\sin (θ)$	$\cos (θ)$	$\tan (θ)$
$\sin (0 °) = 0$	$\cos (0 °) = 1$	$\tan (0 °) = \frac{0}{1} = 0$
$\sin (30 °) = \frac{1}{2}$	$\cos (30 °) = \frac{\sqrt{3}}{2}$	$\tan (30 °) = \frac{1}{2} \cdot \frac{2}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
$\sin (45 °) = \frac{\sqrt{2}}{2}$	$\cos (45 °) = \frac{\sqrt{2}}{2}$	$\tan (45 °) = \frac{\sqrt{2}}{2} \cdot \frac{2}{\sqrt{2}} = 1$
$\sin (60 °) = \frac{\sqrt{3}}{2}$	$\cos (60 °) = \frac{1}{2}$	$\tan (60 °) = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3}$
$\sin (90 °) = 1$	$\cos (90 °) = 0$	$\tan (90 °) = \frac{1}{0} = undef .$

Note that:

for angles with their terminal arm in Quadrant II, since sine is positive and cosine is negative, tangent is negative.
for angles with their terminal arm in Quadrant III, since sine is negative and cosine is negative, tangent is positive.
for angles with their terminal arm in Quadrant IV, since sine is negative and cosine is positive, tangent is negative.

You can plot these points on a coordinate plane to show part of the function, the part between $0$ and $2 π$ .

For values of $θ$ less than $0$ or greater than $2 π$ you can find the value of $θ$ using the reference angle .

The graph of the function over a wider interval is shown below.

Note that the domain of the function is the whole real line and the range is $- \infty \leq y \leq \infty$ .

Tangent Function

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