In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. tan⁻¹ (−20)

Verified step by step guidance

1. Recognize that the expression \( \tan^{-1}(-20) \) represents the inverse tangent (arctangent) function, which gives the angle whose tangent is \(-20\).
2. Recall that the range of the inverse tangent function is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) radians or \( (-90^\circ, 90^\circ) \) degrees, so the result will be an angle in this interval.
3. Use a calculator set to the desired angle mode (degrees or radians) to evaluate \( \tan^{-1}(-20) \).
4. Input the value \(-20\) into the inverse tangent function on the calculator, typically by pressing the \( \tan^{-1} \) or \( \arctan \) button followed by \(-20\).
5. Round the resulting angle to two decimal places as required.

Key Concepts

Inverse Tangent Function (arctan)

The inverse tangent function, denoted as tan⁻¹ or arctan, returns the angle whose tangent is a given number. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known. The output angle is typically in radians or degrees within the range of -90° to 90° (or -π/2 to π/2).

Using a Calculator for Inverse Trigonometric Functions

Calculators have specific modes (degree or radian) that affect the output of inverse trig functions. To find tan⁻¹(−20), ensure the calculator is set to the correct mode, then input the value to get the angle. The result should be rounded to the required decimal places, here two decimals.

Interpreting Negative Inputs in Inverse Tangent

A negative input to tan⁻¹ indicates the angle lies in the fourth or second quadrant, depending on the function's range. Since arctan outputs angles between -90° and 90°, a negative input results in a negative angle, reflecting the direction below the x-axis. Understanding this helps interpret the angle's sign and position.