Inverse Cosine

Definition

The inverse cosine, also known as the arccosine, is the inverse function of the cosine trigonometric function. It is used to find the angle whose cosine is a given value, allowing for the determination of the angle from the ratio of the adjacent and hypotenuse sides of a right triangle.

5 Must Know Facts For Your Next Test

1. The inverse cosine function is denoted as $\cos^{-1}(x)$ or $\arccos(x)$, where $x$ is the given value of the cosine.
2. The inverse cosine function returns an angle between 0 and 180 degrees (or 0 and $\pi$ radians) whose cosine is the given value.
3. The inverse cosine function is used to find the angle in a right triangle when the ratio of the adjacent side to the hypotenuse (the cosine) is known.
4. Integrals involving the inverse cosine function can arise in calculus when integrating functions that contain the cosine function.
5. The inverse cosine function is one of the four basic inverse trigonometric functions, along with inverse sine, inverse tangent, and inverse cotangent.

Review Questions

• Explain how the inverse cosine function is used to determine the angle in a right triangle given the ratio of the adjacent side to the hypotenuse.
  • The inverse cosine function, $\cos^{-1}(x)$ or $\arccos(x)$, is used to find the angle in a right triangle when the ratio of the adjacent side to the hypotenuse (the cosine) is known. By inputting the given cosine value into the inverse cosine function, the angle whose cosine is that value can be determined. This allows for the calculation of the unknown angle in the right triangle, which is crucial for solving various geometric and trigonometric problems.
• Describe how the inverse cosine function is related to the integration of functions containing the cosine function.
  • In the context of integrals resulting in inverse trigonometric functions, the inverse cosine function can arise when integrating functions that contain the cosine function. The integration of expressions involving the cosine function may lead to the inverse cosine function, $\cos^{-1}(x)$ or $\arccos(x)$, as the result. This relationship between the cosine function and its inverse allows for the evaluation of certain integrals that would not be possible without the use of inverse trigonometric functions.
• Analyze the properties of the inverse cosine function and explain how it differs from the original cosine function.
  • The inverse cosine function, $\cos^{-1}(x)$ or $\arccos(x)$, has several key properties that distinguish it from the original cosine function. Firstly, the domain of the inverse cosine function is limited to the range $[-1, 1]$, as the cosine function is only defined within this interval. Additionally, the inverse cosine function returns an angle between 0 and 180 degrees (or 0 and $\pi$ radians), whereas the original cosine function can return angles in all four quadrants of the unit circle. Furthermore, the inverse cosine function is not one-to-one, meaning that multiple angles can have the same cosine value, requiring the function to return a specific angle within the defined range.

Related terms

Inverse Trigonometric Functions:

Inverse trigonometric functions are used to find the angle given the value of a trigonometric ratio, such as sine, cosine, or tangent.

Right Triangle:

A right triangle is a triangle in which one of the angles is a 90-degree angle, or a right angle.

Trigonometric Ratios: Trigonometric ratios, such as sine, cosine, and tangent, represent the relationships between the sides of a right triangle.