Inverse Trigo Function

Definition of Inverse Trigonometric Functions

Every mathematical function, from the simplest to the most complex, has an inverse. In mathematics, inverse usually means opposite. For addition, the inverse is subtraction. For multiplication, it's division. And for trigonometric functions, it's the inverse trigonometric functions.

Trigonometric functions are the functions of an angle. The term function is used to describe the relationship between two sets of numbers or variables. In modern mathematics, there are six basic trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent. The inverse of these functions are inverse sine, inverse cosine, inverse tangent, inverse secant, inverse cosecant, and inverse cotangent.

Trigonometric Ratios

The trigonometric functions can all be defined as ratios of the sides of a right triangle. Since all right triangles conform to the Pythagorean Theorem, as long as the angles of two right triangles are the same, their sides will be proportional. Because of this, the ratios of one side to another will always be the same. Take a look at this example.


These triangles have the same angle measures, so their sides are proportional. Any ratio of one side to another will be the same for both triangles.

6/10 = 3/5

By discovering that these ratios are the same for any sized right triangle (as long as they have the same angle measure), the trigonometric functions were discovered. These functions relate one angle of a triangle to the ratio of two of its sides.




Because of these ratios, when an angle (other than the right angle) of a right triangle and at least one side are known, you can determine the length of the other sides using these ratios. And inversely, when the lengths of two sides are known, the angle measure can be determined.
Since memorizing these ratios can prove to be difficult, there is a mnemonic that helps keep them straight. SOH CAH TOA is a helpful device to remember which ratio goes with which function.
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent

Inverse Trigonometric Functions

The inverse trigonometric functions are used to determine the angle measure when at least two sides of a right triangle are known. The particular function that should be used depends on what two sides are known. For example, if you know the hypotenuse and the side opposite the angle in question, you could use the inverse sine function. If you know the side opposite and the side adjacent to the angle in question, the inverse tangent is the function you need.

There are two methods for determining an inverse trigonometric function. The first is by using a table containing all the results for every ratio. It can be tedious and cumbersome. The other is using a scientific calculator. The inverse functions for the sine, cosine, and tangent can be determined quickly.

These inverse functions have practical uses in navigation, physics, engineering, and other sciences.

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