교과서 질문
In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + cos 2x
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
모든 교과서
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 65
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 652장, 문제 65
In Exercises 65–66, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in centimeters. In each exercise, find: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 20 cos π/4 t
검증된 단계별 안내1
Identify the given equation of simple harmonic motion: \(d = 20 \cos \left( \frac{\pi}{4} t \right)\), where \(d\) is displacement in centimeters and \(t\) is time in seconds.
To find the maximum displacement, recognize that the amplitude of the motion is the coefficient in front of the cosine function. So, the maximum displacement is the absolute value of this coefficient.
To find the frequency, recall that the general form of simple harmonic motion is \(d = A \cos(\omega t)\), where \(\omega\) is the angular frequency in radians per second. Identify \(\omega\) from the equation and use the relation \(f = \frac{\omega}{2\pi}\) to find the frequency in hertz (cycles per second).
To find the time required for one cycle (the period), use the formula \(T = \frac{1}{f}\), where \(T\) is the period in seconds and \(f\) is the frequency found in the previous step.
Summarize the results: maximum displacement is the amplitude, frequency is calculated from angular frequency, and period is the reciprocal of frequency.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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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 displacement varies sinusoidally with time, typically modeled by equations like d = A cos(ωt), where A is amplitude and ω is angular frequency.
추천 영상:
04:46
Products of Complex Numbers in Polar Form
Amplitude and Maximum Displacement
Amplitude is the maximum displacement from the equilibrium position in SHM. It represents the peak value of the cosine or sine function in the motion equation, indicating how far the object moves from its central position.
추천 영상:
5:05Amplitude and Reflection of Sine and Cosine
Frequency and Period of Oscillation
Frequency is the number of complete oscillations per second, measured in hertz (Hz), and is related to angular frequency ω by f = ω/(2π). The period is the time for one full cycle, given by T = 1/f, representing how long the object takes to repeat its motion.
추천 영상:
5:33Period of Sine and Cosine Functions
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교과서 질문
In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + sin 2x
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