Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 1 - Functions

Problema 82

Capitolo 1, Problema 82

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

Guida verificata passo dopo passo

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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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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06:35

Derivatives of Other Inverse Trigonometric Functions

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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3:48

Evaluate Composite Functions - Special Cases

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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Percorso guidato

6:04

Introduction to Trigonometric Functions

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