Domanda del libro di testo
Graph each function over a two-period interval.
y = -2 + (1/2) sin 3x
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 5, Problema 61
Consider the following function from Example 5. Work these exercises in order.
y = -2 - cot (x - π/4)
Use the fact that the period of this function is π to find the next positive x-intercept. Round to the nearest hundredth.
Guida verificata passo dopo passo1
Identify the given function: \(y = -2 - \cot\left(x - \frac{\pi}{4}\right)\). We want to find the x-intercepts, where \(y = 0\).
Set the function equal to zero to find the x-intercepts: \(0 = -2 - \cot\left(x - \frac{\pi}{4}\right)\).
Rearrange the equation to isolate the cotangent term: \(\cot\left(x - \frac{\pi}{4}\right) = -2\).
Recall that the cotangent function has a period of \(\pi\), so the general solution for \(\cot \theta = -2\) is \(\theta = \cot^{-1}(-2) + k\pi\), where \(k\) is any integer.
Substitute back \(\theta = x - \frac{\pi}{4}\) and solve for \(x\): \(x = \cot^{-1}(-2) + \frac{\pi}{4} + k\pi\). Use the smallest positive \(x\)-intercept found (for some integer \(k\)) and then add the period \(\pi\) to find the next positive x-intercept. Round your answer to the nearest hundredth.
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Cotangent Function and Its Properties
The cotangent function, cot(x), is the reciprocal of the tangent function and is defined as cos(x)/sin(x). It has vertical asymptotes where sin(x) = 0 and zeros where cos(x) = 0. Understanding its behavior and graph is essential for identifying intercepts and transformations.
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Percorso guidato
5:37
Introduction to Cotangent Graph
Period of Trigonometric Functions
The period of a trigonometric function is the length of one complete cycle before the function repeats. For cotangent, the standard period is π. When the function is transformed, such as cot(x - π/4), the period remains π, which helps in finding subsequent intercepts by adding multiples of the period.
Video consigliato:
Percorso guidato
5:33Period of Sine and Cosine Functions
Finding X-Intercepts of Transformed Functions
X-intercepts occur where the function equals zero. For y = -2 - cot(x - π/4), set y = 0 and solve for x. This involves isolating cot(x - π/4) and using the periodicity to find the next positive solution, then rounding the result as required.
Video consigliato:
Percorso guidato
4:22Domain and Range of Function Transformations
Pratica correlata
Domanda del libro di testo
Graph each function over a two-period interval.
y = 1 - 2 cos ((1/2)x)
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Domanda del libro di testo
Graph each function over a two-period interval.
y = sin [2(x + π/4) ] + 1/2
875
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Domanda del libro di testo
Consider the following function from Example 5. Work these exercises in order.
y = -2 - cot (x - π/4)
Based on the answer in Exercise 58 and the fact that the cotangent function has period π, give the general form of the equations of the asymptotes of the graph of y = -2 - cot (x - π/4).
Let n represent any integer.
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