교과서 질문
Fill in the blank(s) to correctly complete each sentence.
The graph of y = sin (x + π/4) is obtained by shifting the graph of y = sin x ______ unit(s) to the ________ (right/left).
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- 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
5장, 문제 2
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the period of this motion?
검증된 단계별 안내1
Identify the general form of the position function for simple harmonic motion, which is usually given by \(s(t) = A \cos(\omega t)\) or \(s(t) = A \sin(\omega t)\), where \(\omega\) is the angular frequency in radians per second.
From the given function \(s(t) = 5 \cos 2t\), recognize that the angular frequency \(\omega\) is 2.
Recall the formula that relates the period \(T\) of the motion to the angular frequency: \(T = \frac{2\pi}{\omega}\).
Substitute the value of \(\omega = 2\) into the formula to express the period as \(T = \frac{2\pi}{2}\).
Simplify the expression to find the period \(T\) in seconds, which represents the time it takes for one complete cycle of the motion.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Simple Harmonic Motion (SHM)
Simple Harmonic Motion describes oscillatory motion where the restoring force is proportional to displacement and acts in the opposite direction. The position function is typically sinusoidal, such as s(t) = A cos(ωt) or s(t) = A sin(ωt), where A is amplitude and ω is angular frequency.
추천 영상:
04:46
Products of Complex Numbers in Polar Form
Angular Frequency (ω)
Angular frequency ω represents how quickly the object oscillates in radians per second. It is the coefficient of t inside the cosine or sine function in SHM equations. The angular frequency relates to the period by the formula ω = 2π / T, where T is the period.
추천 영상:
3:47Introduction to Common Polar Equations
Period of Oscillation (T)
The period T is the time it takes for one complete cycle of motion. It is inversely related to angular frequency by T = 2π / ω. Knowing ω from the position function allows calculation of T, which answers how long one full oscillation lasts.
추천 영상:
5:33Period of Sine and Cosine Functions
관련 실천
교과서 질문
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the frequency?
704
views
교과서 질문
Fill in the blank(s) to correctly complete each sentence.
The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x ______ unit(s) to the ________ (right/left).
769
views
교과서 질문
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = ½ cos π x
2
1039
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교과서 질문
An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the amplitude of this motion?
700
views
교과서 질문
Fill in the blank(s) to correctly complete each sentence.
The graph of y = 4 sin x is obtained by stretching the graph of y = sin x vertically by a factor of ________.
742
views