Multiple Choice
Evaluate the expression.
0. Functions
Inverse Trigonometric Functions
1:21 minutesProblem 82
Textbook Question
Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
Verified step by step guidance1
Understand the problem: We need to evaluate \( \tan(\tan^{-1}(1)) \). This involves the inverse trigonometric function \( \tan^{-1} \), which is the inverse of the tangent function.
Recall the definition of \( \tan^{-1}(x) \): It is the angle \( \theta \) such that \( \tan(\theta) = x \) and \( \theta \) is in the range \(-\frac{\pi}{2} < \theta < \frac{\pi}{2} \).
Apply the definition to \( \tan^{-1}(1) \): We need to find an angle \( \theta \) such that \( \tan(\theta) = 1 \). The angle \( \theta \) that satisfies this within the range is \( \frac{\pi}{4} \), because \( \tan(\frac{\pi}{4}) = 1 \).
Substitute back into the original expression: Now that we know \( \tan^{-1}(1) = \frac{\pi}{4} \), we substitute this into the expression to get \( \tan(\frac{\pi}{4}) \).
Evaluate \( \tan(\frac{\pi}{4}) \): Since \( \tan(\frac{\pi}{4}) = 1 \), the expression \( \tan(\tan^{-1}(1)) \) simplifies to 1.
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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 an^{-1}(x), are used to find the angle whose tangent is x. These functions essentially reverse the action of the standard trigonometric functions. For example, if an( heta) = x, then heta = an^{-1}(x). Understanding these functions is crucial for evaluating expressions involving angles and their corresponding trigonometric ratios.
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Function Composition
Function composition involves applying one function to the result of another function. In the expression an( an^{-1}(1)), the an^{-1}(1) computes the angle whose tangent is 1, which is rac{ ext{π}}{4} radians. Then, applying the tangent function to this angle returns the original input, demonstrating that an( an^{-1}(x)) = x for all x in the domain of the inverse function.
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Trigonometric Values
Trigonometric values are the outputs of trigonometric functions for specific angles. For instance, an( rac{ ext{π}}{4}) equals 1, as the tangent of 45 degrees (or rac{ ext{π}}{4} radians) is 1. Knowing the standard values of trigonometric functions for common angles (0, rac{ ext{π}}{6}, rac{ ext{π}}{4}, rac{ ext{π}}{3}, and rac{ ext{π}}{2}) is essential for quickly evaluating expressions without a calculator.
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