Textbook Question
Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹ (1/x ) = π/6
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.51
Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = arccos (-0.39876459)
Verified step by step guidance1
Understand that \( \theta = \arccos(-0.39876459) \) means you are looking for an angle \( \theta \) whose cosine is \(-0.39876459\).
Recall that the range of the \( \arccos \) function is from \( 0 \) to \( 180 \) degrees, meaning the angle \( \theta \) will be within this range.
Use a calculator with an inverse cosine function to find \( \theta \). Ensure the calculator is set to degree mode to get the result in decimal degrees.
Input \(-0.39876459\) into the calculator and apply the \( \arccos \) function to find the angle \( \theta \).
The calculator will provide the angle \( \theta \) in decimal degrees, which is the solution to the problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as arccos, are used to find the angle whose cosine is a given value. For example, if θ = arccos(x), then cos(θ) = x. These functions are essential for solving problems where the angle is unknown and can be found using the ratio of sides in a right triangle.
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Cosine Function
The cosine function relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is a fundamental concept in trigonometry, and its values range from -1 to 1. Understanding how to interpret these values is crucial when using inverse functions to find angles.
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Calculator Functions
Most scientific calculators have specific functions for calculating inverse trigonometric values. When using a calculator to find arccos(-0.39876459), it is important to ensure that the calculator is set to the correct mode (degrees or radians) to obtain the desired output. Familiarity with these functions enhances accuracy in solving trigonometric problems.
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