Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. cos⁻¹ (− 1/2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
1:49 minutesProblem 37
Textbook Question
In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. ___ tan⁻¹ (−√473)
Verified step by step guidance1
Recognize that the expression involves the inverse tangent function, written as \(\tan^{-1}(x)\), which gives the angle whose tangent is \(x\).
Identify the value inside the inverse tangent function: \(-\sqrt{473}\). This is a negative number, so the angle will be in either the second or fourth quadrant, depending on the range of \(\tan^{-1}\).
Recall that the principal value of \(\tan^{-1}(x)\) lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) (or between -90° and 90°), so the angle will be negative because the input is negative.
Use a calculator to find the angle in radians or degrees by entering \(\tan^{-1}(-\sqrt{473})\). Make sure your calculator is set to the correct mode (degrees or radians) as required.
Round the resulting angle to two decimal places as the problem requests.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Tangent Function (tan⁻¹ or 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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Inverse Tangent
Evaluating Square Roots and Negative Values
Understanding how to simplify and interpret square roots, especially when combined with negative signs, is essential. Here, the expression involves the negative of the square root of 473, which is a negative real number. Recognizing this helps in correctly inputting the value into a calculator for the inverse tangent function.
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Using a Calculator for Trigonometric Functions
Calculators can compute inverse trigonometric functions and provide decimal approximations. It is important to ensure the calculator is set to the correct angle mode (degrees or radians) and to round the final answer to the specified decimal places, in this case, two decimal places.
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