What is the amplitude of the sinusoid given by [tex]y=-3 \sin (5 x) [/tex]?
Answer (1)
Identify the sinusoidal function: y = − 3 sin ( 5 x ) .
Recall that the amplitude is the absolute value of the coefficient of the sine function.
Calculate the absolute value: ∣ − 3∣ = 3 .
State the amplitude: 3 .
Explanation
Understanding the Problem We are given the sinusoidal function y = − 3 sin ( 5 x ) . Our goal is to find the amplitude of this sinusoid. Recall that the general form of a sinusoidal function is given by y = A sin ( B x + C ) + D , where ∣ A ∣ represents the amplitude, B affects the period, C affects the phase shift, and D affects the vertical shift.
Identifying the Amplitude In our given equation, y = − 3 sin ( 5 x ) , we can identify the corresponding values as A = − 3 , B = 5 , C = 0 , and D = 0 . The amplitude is the absolute value of the coefficient A .
Calculating the Amplitude To find the amplitude, we take the absolute value of A , which is − 3 . The absolute value of − 3 is ∣ − 3∣ = 3 . Therefore, the amplitude of the sinusoid is 3.
Final Answer The amplitude of the sinusoid y = − 3 sin ( 5 x ) is 3.
Examples
Understanding the amplitude of a sinusoidal function is crucial in many real-world applications. For example, in acoustics, the amplitude of a sound wave determines its loudness. In electrical engineering, the amplitude of an alternating current (AC) signal represents the maximum voltage or current. In seismology, the amplitude of a seismic wave indicates the intensity of an earthquake. By analyzing the amplitude, we can quantify the strength or intensity of these phenomena.
