Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = arccos (-0.39876459)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.57
Textbook Question
Solve each equation for x.
arccos x + arctan 1 = 11π/12
Verified step by step guidance1
Recognize that \( \arctan(1) \) is a known angle. Since \( \tan(\frac{\pi}{4}) = 1 \), we have \( \arctan(1) = \frac{\pi}{4} \).
Substitute \( \arctan(1) = \frac{\pi}{4} \) into the equation: \( \arccos(x) + \frac{\pi}{4} = \frac{11\pi}{12} \).
Isolate \( \arccos(x) \) by subtracting \( \frac{\pi}{4} \) from both sides: \( \arccos(x) = \frac{11\pi}{12} - \frac{\pi}{4} \).
Find a common denominator to simplify the right side: \( \frac{11\pi}{12} - \frac{3\pi}{12} = \frac{8\pi}{12} \).
Simplify \( \frac{8\pi}{12} \) to \( \frac{2\pi}{3} \), so \( \arccos(x) = \frac{2\pi}{3} \). Solve for \( x \) using the cosine function: \( x = \cos(\frac{2\pi}{3}) \).
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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 and arctan, are used to find angles when given a ratio of sides in a right triangle. For example, arccos x gives the angle whose cosine is x, while arctan 1 gives the angle whose tangent is 1, which is π/4. Understanding these functions is crucial for solving equations involving angles.
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Angle Addition
The concept of angle addition is essential when working with trigonometric equations. In this case, the equation involves the sum of two angles: arccos x and arctan 1. Recognizing that arctan 1 equals π/4 allows us to rewrite the equation as arccos x + π/4 = 11π/12, facilitating the isolation of x.
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Solving Trigonometric Equations
Solving trigonometric equations often involves isolating the variable and using known values of trigonometric functions. In this problem, after simplifying the equation, we can find x by applying the cosine function to both sides. This process requires familiarity with the unit circle and the properties of trigonometric functions to determine valid solutions.
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