Textbook Question
Evaluate each expression without using a calculator.
tan⁻¹ (tan (π/4))
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 3
Textbook Question
Which one of the following equations has solution 3π/4
a. arctan 1 = x
b. arcsin √2/2 = x
c. arccos (―√2 /2) = x
Verified step by step guidance1
Recall that the inverse trigonometric functions arctan, arcsin, and arccos return principal values within specific ranges: arctan returns values in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), arcsin returns values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and arccos returns values in \([0, \pi]\).
Check if \(x = \frac{3\pi}{4}\) lies within the principal value range of each inverse function: for arctan, \(\frac{3\pi}{4}\) is outside \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\); for arcsin, \(\frac{3\pi}{4}\) is outside \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\); for arccos, \(\frac{3\pi}{4}\) is inside \([0, \pi]\).
Evaluate the trigonometric values at \(x = \frac{3\pi}{4}\): \(\tan\left(\frac{3\pi}{4}\right)\), \(\sin\left(\frac{3\pi}{4}\right)\), and \(\cos\left(\frac{3\pi}{4}\right)\) to see which matches the given values in the equations.
Compare the given values in the problem with the values from step 3: \(\tan\left(\frac{3\pi}{4}\right)\), \(\sin\left(\frac{3\pi}{4}\right)\), and \(\cos\left(\frac{3\pi}{4}\right)\) to identify which equation has \(x = \frac{3\pi}{4}\) as a solution.
Conclude which inverse trigonometric equation corresponds to \(x = \frac{3\pi}{4}\) based on the principal value ranges and the matching trigonometric values.
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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 arcsin, arccos, and arctan, return the angle whose trigonometric ratio matches a given value. They are used to find angles from known sine, cosine, or tangent values, with outputs restricted to specific principal value ranges.
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Principal Value Ranges of Inverse Functions
Each inverse trig function has a principal value range where it produces unique outputs: arcsin ranges from -π/2 to π/2, arccos from 0 to π, and arctan from -π/2 to π/2. Understanding these ranges helps determine if a given angle like 3π/4 can be a solution.
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Evaluating Trigonometric Values at Specific Angles
Knowing the exact sine, cosine, and tangent values at common angles such as π/4, 3π/4, and π/2 is essential. For example, sin(3π/4) = √2/2, cos(3π/4) = -√2/2, and tan(3π/4) = -1, which helps identify which inverse function equation corresponds to the angle 3π/4.
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