Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 82

Chapter 1, Problem 82

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$$\tan\]\left$($\tan$^{-1}1$\right$)$

Verified step by step guidance

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.

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.

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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