What is the period of the sinusoid given by [tex]y=-3 \cos \left(\frac{2 \pi}{5} x\right) ?[/tex]
Answer (2)
Identify the coefficient of x in the cosine function: B = 5 2 π .
Apply the formula for the period of a cosine function: Period = ∣ B ∣ 2 π .
Substitute the value of B into the formula: Period = 5 2 π 2 π .
Simplify the expression to find the period: 5 .
Explanation
Understanding the Problem We are asked to find the period of the sinusoidal function y = − 3 cos ( 5 2 π x ) . The general form of a cosine function is y = A cos ( B x + C ) + D , where the period is given by ∣ B ∣ 2 π . In our case, B = 5 2 π .
Applying the Period Formula The period of the given sinusoid is calculated using the formula: Period = ∣ B ∣ 2 π .We have B = 5 2 π , so we substitute this value into the formula: Period = ∣ 5 2 π ∣ 2 π = 5 2 π 2 π .
Calculating the Period To simplify the expression, we multiply the numerator and denominator by 5: Period = 5 2 π 2 π = 2 π 2 π × 5 = 5 .Therefore, the period of the given sinusoid is 5.
Final Answer The period of the sinusoid y = − 3 cos ( 5 2 π x ) is 5.
Examples
Understanding the period of a sinusoidal function is crucial in many real-world applications, such as analyzing sound waves or alternating current (AC) circuits. For example, if you are designing an audio system, knowing the period of a sound wave helps you determine the frequency and pitch of the sound. Similarly, in electrical engineering, the period of an AC signal is essential for designing circuits and understanding their behavior. By calculating the period, engineers can ensure that systems operate correctly and efficiently.
The period of the sinusoid y = − 3 cos ( 5 2 π x ) is 5. This is calculated using the period formula for cosine functions. Thus, the wave will repeat every 5 units along the x-axis.
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