key term - Tan^-1

Definition

The inverse tangent function, denoted as tan^-1 or arctan, is a trigonometric function that calculates the angle whose tangent is a given value. It is used to find the angle when the tangent ratio is known, reversing the process of the tangent function.

5 Must Know Facts For Your Next Test

1. The tan^-1 function returns the angle (in radians) whose tangent is the given value, whereas the tangent function finds the tangent ratio given an angle.
2. The domain of the tan^-1 function is the set of all real numbers, and the range is $(-\pi/2, \pi/2)$, which corresponds to the angles where the tangent is defined.
3. The tan^-1 function is useful in solving right triangle problems, where the tangent ratio is known, and the angle needs to be determined.
4. When working with inverse trigonometric functions, it is important to consider the appropriate quadrant of the angle based on the sign of the trigonometric ratio.
5. The tan^-1 function can be evaluated using a calculator or by referring to a table of trigonometric values.

Review Questions

• Explain how the tan^-1 function is used to solve right triangle problems.
  • In right triangle problems, the tan^-1 function is used to find the angle when the tangent ratio is known. For example, if the tangent ratio of an angle is 0.5, the tan^-1 function can be used to find the angle, which would be approximately 26.57 degrees or 0.463 radians. This is useful when the side lengths of a right triangle are given, and the angle needs to be determined using the tangent ratio.
• Describe the relationship between the tan^-1 function and the tangent function.
  • The tan^-1 function is the inverse of the tangent function. Whereas the tangent function finds the tangent ratio given an angle, the tan^-1 function finds the angle given the tangent ratio. This inverse relationship means that if $y = \tan^-1(x)$, then $\tan(y) = x$. In other words, the tan^-1 function undoes or reverses the operation of the tangent function, allowing you to determine the angle when the tangent ratio is known.
• Analyze the significance of the domain and range of the tan^-1 function in the context of inverse trigonometric functions.
  • The domain and range of the tan^-1 function are important considerations when working with inverse trigonometric functions. The domain of tan^-1 is the set of all real numbers, meaning the function can accept any real number as input. However, the range of tan^-1 is $(-\pi/2, \pi/2)$, which corresponds to the angles where the tangent function is defined. This range restriction is crucial, as it ensures that the tan^-1 function returns a unique angle for each input value. The limited range also reflects the fact that the tangent function is periodic, with a period of $\pi$, and the tan^-1 function must account for this periodicity to provide the appropriate angle solution.

Related terms

Inverse Trigonometric Functions:

Inverse trigonometric functions, such as sin^-1, cos^-1, and tan^-1, are used to find the angle when a trigonometric ratio is known, essentially undoing or reversing the original trigonometric function.

Trigonometric Ratios:

The trigonometric ratios, including sine, cosine, and tangent, relate the sides of a right triangle to the angles within the triangle.

Radian Measure:

Radian measure is a way of expressing angles, where one full revolution is equal to $2\pi$ radians, as opposed to the more familiar degree measure.