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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 44
Capitolo 1, Problema 44

Solve the following equations.
tan22θ=1,0<θ<π\(\tan\)^22\(\theta\)=1,0\(\lt{\theta}\)\(\lt{\pi}\)

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1
Recognize that the equation \( \tan^2(2\theta) = 1 \) implies \( \tan(2\theta) = \pm 1 \).
Consider the principal values for \( \tan(2\theta) = 1 \), which occur at \( 2\theta = \frac{\pi}{4} + n\pi \) for integer \( n \).
Consider the principal values for \( \tan(2\theta) = -1 \), which occur at \( 2\theta = \frac{3\pi}{4} + n\pi \) for integer \( n \).
Solve for \( \theta \) by dividing each equation by 2: \( \theta = \frac{\pi}{8} + \frac{n\pi}{2} \) and \( \theta = \frac{3\pi}{8} + \frac{n\pi}{2} \).
Determine the values of \( \theta \) that satisfy \( 0 < \theta < \pi \) by substituting different integer values for \( n \) and checking the range.

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The tangent function, specifically, is defined as the ratio of the opposite side to the adjacent side. Understanding these functions is crucial for solving equations involving angles, as they provide the necessary relationships to manipulate and solve for unknowns.
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Introduction to Trigonometric Functions

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, allow us to find angles when given a ratio. For example, if we know that an(θ) = 1, we can use the inverse tangent function to determine that θ = π/4. This concept is essential for solving trigonometric equations, as it helps to isolate the angle variable.
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Derivatives of Other Inverse Trigonometric Functions

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For instance, the tangent function has a period of π, which means that an(θ) = an(θ + nπ) for any integer n. This property is important when solving equations over specific intervals, as it allows us to find all possible solutions within the given range.
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Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo

Solve the following equations.


{Use of Tech} sin2θ=15,0<θ<π2\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\)\(\lt{\frac{\pi}{2}\)}

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Solve the following equations.

sin3x=22,0≤x<2π\(\sin\)3x=\(\frac{\sqrt{2}\)}{2},0\(\leq{x}\)\(\lt{2\pi}\)

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More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.


ƒ o g

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Domanda del libro di testo

Slope functions Determine the slope function S (x) for the following functions

ƒ(x)=∣x∣ƒ(x) = | x |

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Domanda del libro di testo

Solve the following equations.

cos3x=sin3x,0≤x<2π\(\cos\)3x=\(\sin\)3x,0\(\leq{x}\)\(\lt\)2\(\pi\)

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Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

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