Tangent (of an angle)
The tangent of an angle is the trigonometric ratio between the adjacent side and the opposite side of a right triangle containing that angle.
Example:
In the triangle shown, 
and 
.
The tangent ratio is the same regardless of the size of the right triangle. So, it is often easiest to consider a right triangle with a hypotenuse of length 1.
The tangent ratio can also be thought of as a function, which takes different values depending on the measure of the angle. You can measure an angle in degrees or radians.
Suppose you have an angle measuring θ radians. 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 s-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 |
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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.
