*Jennifer Beddoe*Show bio

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

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Instructor:
*Jennifer Beddoe*
Show bio

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Arccosine is an inverse trigonometric function. It happens to be the inverse of cosine and can be used to solve more complicated right angle problems. This lesson will define the arccosine in more detail and give some example problems. Be prepared for a quiz at the end of the lesson.
Updated: 08/31/2021

In mathematics, most operations have an **inverse operation**, which is the opposite of the original operation. For example, subtraction is the inverse of addition, and multiplication is the opposite of division.

Trigonometric functions are no different. They all have inverse operations. Here is a table of the functions and inverse functions:

Function | Inverse Function |
---|---|

sine | arcsine |

cosine | arccosine |

tangent | arctangent |

secant | arcsecant |

cosecant | arccosecant |

cotangent | arccotangent |

Inverse trigonometric functions have practical applications in physics, engineering, and navigation.

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The **cosine (cos)** of an angle in a right triangle is a trigonometric function that can be defined as the ratio of the side adjacent to the angle and the hypotenuse.

The graph of the cosine function looks like this:

The cosine function can be used to solve problems involving a right triangle. For example: Say there is a ladder leaning against the side of a building. You want to know how tall the ladder is without moving it or climbing it. You know that the base of the ladder is 3 feet from the building and that the angle the ladder makes with the ground is 75°. You can use the cosine function to determine the length of the ladder.

cos(*x*) = adjacent / hypotenuse

cos(75) = 3ft / hypotenuse

hypotenuse = 3 ft / cos75

hypotenuse = 3 / 0.25882

hypotenuse = 11.59 ft

So the ladder is 11.59 ft tall.

The **arccosine** is the inverse of the cosine. It's most useful when trying to find the angle measure when two sides of a triangle are known.

The graph of the arccosine looks like this:

To solve problems using the arccosine, you need a scientific calculator. It should have a button, usually above the cosine button that looks like this:

The arccosine allows you to find the measure of an angle when you know the ratio of the adjacent side to the hypotenuse.

Let's try a few examples.

1) Find the angle measure Î in the following triangle:

cos(Î) = 7 / 13

cos(Î) = 0.538

Finding the arccosine, we get:

Î = 57.4°

2) Let's look at the ladder example from before. If you knew that the ladder was 7 feet tall and that it was 3 feet away from the wall at the ground, you could determine the angle that the ladder made with the ground using the arccosine:

cos(*x*) = 3 feet / 7 feet (adjacent / hypotenuse)

cos (*x*) = 0.4286

Now, to the find the measure of that angle, simply find the arccosine of 0.4286:

*x* = 64.6°

As with any other inverse operations, you can always check your answer by doing the opposite.

Check this answer by calculating cos(64.6°):

cos(64.6) = 0.4286

So the angle that the ladder makes with the ground is 64.6°.

An **inverse operation** is the opposite of the original operation. The **cosine (cos)** of an angle in a right triangle is the ratio of the side adjacent to the angle and the hypotenuse. The **arccosine** is the inverse function of the cosine function. This means that they are opposite functions, and one will cancel out the other. The arccosine is mainly used to determine the measure of an angle when two sides of a right triangle are known. It has applications in navigation, engineering, and other sciences.

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