# Inverse Trigonometric Functions: Definition & Problems

## What Are Inverse Trig Functions?

Every mathematical function, from the simplest to the most complex, has an **inverse**, or 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 trigonometric 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 the 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 Trig 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.

## Example Problems

Let's try out some examples.

1.) In this triangle, solve for *x*:

Since we know the side opposite angle *x* and the hypotenuse, we can use the inverse sine function to determine the angle measure of *x*:

sin(*x*) = Opposite / Hypotenuse

sin(*x*) = 8 / 11

sin(*x*) = 0.73

*x* = 47°

2.) Now solve for *x* in this triangle:

For this problem, we know the sides opposite and adjacent to the angle in question. This time we will use the inverse tangent to solve for the angle measure:

tan(*x*) = Opposite / Adjacent

tan(*x*) = 7 / 5

tan(*x*) = 1.4

*x* = 54°

3.) Say you drive 24 miles up a hill, and in that distance travel only 6 horizontal miles. At what angle of incline are you ascending (angle *x*)?

This time, you will use the inverse of the cosine function since you know the side adjacent to the angle and the hypotenuse:

cos(*x*) = Adjacent / Hypotenuse

cos(*x*) = 6 / 24

cos(*x*) = 0.25

*x* = 76°

## Lesson Summary

Every mathematical function has an **inverse**, or opposite. **Trigonometric functions** are the functions of an angle. There are six basic trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent. The **inverse trigonometric functions** of these are inverse sine, inverse cosine, inverse tangent, inverse secant, inverse cosecant, and inverse cotangent.

When two of the three side lengths of a right triangle are known, the inverse trigonometric function is used to find the measure of an angle. **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 are useful in engineering, navigation, building, and other scientific fields.

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## Practice Problems: Inverse Trigonometric Functions

#### Key Terms

- Inverse Trigonometric Functions: Functions that give the value of the angle in Degrees or Radians from its Trigonometric Function Value.

#### Materials Needed

- Paper
- Pencil
- Scientific Calculator in Degree Mode

#### Practice Problems: Show All Your Work

(a) Triangle ABC has a right angle at C. Next, side AB = 7 m and side CB = 4 m. Using inverse Trigonometric functions, find the degree measures of angle B and angle A.

(b) Triangle EFG has a right angle at G. Next, side EG = 3 in and side GF = 2 in. Using inverse Trigonometric functions, find the degree measures of angle E and angle F.

#### Answers (To Check Your Work):

(a) Below is a sketch of triangle ABC.

Since ABC is a right-angled triangle with a right angle at C, we can use the Trigonometric Functions in a right triangle to get the ratio:

cos B = adjacent/hypotenuse = 4/7.

Applying the inverse cosine function to both sides of the above relation yields:

cos^(-1) (cos B) = cos^(-1) (4/7).

Using a scientific calculator in degree mode gives us:

B = 55.15 degrees.

Next, since the sum of the angles of a triangle is 180 degrees, we get angle A = 180 - (angle B + angle C) = 180 - (55.15 + 90) = 34.85 degrees.

**Angle B = 55.15 degrees and angle A = 34.85 degrees.**

(b) Below is a sketch of triangle EFG.

Since EFG is a right-angled triangle with a right angle at G, we can use the Trigonometric Functions in a right triangle to get the ratio:

tan F = opposite/adjacent = 3/2.

Applying the inverse tangent function to both sides of the above relation yields:

tan^(-1) (tan F) = tan^(-1) (3/2).

Using a scientific calculator in degree mode gives us:

F = 56.31 degrees.

Next, since the sum of the angles of a triangle is 180 degrees, we get angle E = 180 - (angle F + angle G) = 180 - (56.31 + 90) = 33.69 degrees.

**Angle F = 56.31 degrees and angle E = 33.69 degrees.**

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