Textbook Question
Find the exact value of each expression. cos⁻¹ √3/2
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
1:41 minutesProblem 35
Textbook Question
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 guidance1
Recognize that the expression \( \tan^{-1}(-20) \) represents the inverse tangent (arctangent) function, which gives the angle whose tangent is \(-20\).
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.
Use a calculator set to the desired angle mode (degrees or radians) to evaluate \( \tan^{-1}(-20) \).
Input the value \(-20\) into the inverse tangent function on the calculator, typically by pressing the \( \tan^{-1} \) or \( \arctan \) button followed by \(-20\).
Round the resulting angle to two decimal places as required.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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).
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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.
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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.
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