Find the amplitude and period of the function.
[tex]y=\frac{1}{2} \cos 6 x[/tex]
Give the exact values, not decimal approximations.
Answer (1)
Identify the amplitude A from the function y = 2 1 cos 6 x , which is A = 2 1 .
Identify B from the function y = 2 1 cos 6 x , which is B = 6 .
Calculate the period using the formula B 2 π , which gives 6 2 π = 3 π .
State the amplitude and period: Amplitude is 2 1 and Period is 3 π .
Explanation
Understanding the Function We are given the function y = 2 1 cos 6 x and we need to find its amplitude and period. Let's recall the general form of a cosine function, which is y = A cos ( B x ) , where ∣ A ∣ represents the amplitude and the period is given by B 2 π .
Identifying the Amplitude In our given function, y = 2 1 cos 6 x , we can identify the amplitude A and the value of B . By comparing the given function with the general form, we see that A = 2 1 and B = 6 . Therefore, the amplitude is ∣ A ∣ = 2 1 = 2 1 .
Calculating the Period Now, let's calculate the period. The formula for the period is B 2 π . We know that B = 6 , so we substitute this value into the formula: Period = 6 2 π = 3 π .
Final Answer Therefore, the amplitude of the function y = 2 1 cos 6 x is 2 1 and the period is 3 π .
Examples
Understanding the amplitude and period of trigonometric functions is crucial in many real-world applications, such as analyzing sound waves or alternating current (AC) circuits. For instance, in music, the amplitude of a sound wave corresponds to the loudness of the sound, while the period relates to the frequency or pitch. Similarly, in electrical engineering, the amplitude of an AC signal represents the voltage, and the period determines the frequency of the current. By adjusting these parameters, we can control the characteristics of sound and electricity, enabling technologies like audio systems and power grids.
