Which cosine function has maximum of 0.5, a minimum of -0.5, and a period of 2π/3?
y=0.5 cos 3 theta
its d
Well, if the cosine function is getting so much attention with maximums and minimums, I guess it's time to bring out the Cosine Function Diva! She's got a maximum of 0.5, a minimum of -0.5, and a period of 2π/3.
But let me tell you, finding the perfect match for her is quite a task. After brainstorming with the Diva, we've come to the conclusion that the function you're looking for is: Cos(3x).
With the passionate and over-the-top personality of this function, it guarantees those maximums and minimums you're searching for. So, let your hair down and enjoy the show!
To find the cosine function that satisfies these conditions, we need to consider the general equation of a cosine function:
cos(x) = A * cos(B(x-C)) + D
Here, A represents the amplitude, B represents the frequency (or period), C represents a horizontal shift, and D represents a vertical shift.
Given that the function has a maximum of 0.5 and a minimum of -0.5, we can determine the amplitude as half the difference between these values:
Amplitude (A) = (0.5 - (-0.5)) / 2 = 0.5
The period of the function is given as 2π/3. The standard period of a cosine function is 2π, but in this case, the period is shorter. To find B (the frequency), we can use the formula:
B = 2π / period
B = 2π / (2π/3) = 3
The horizontal shift, C, does not affect the maximum, minimum, or period, so we can assume C = 0.
Finally, the vertical shift, D, is the average of the maximum and minimum values:
D = (0.5 + (-0.5)) / 2 = 0
Putting all these values into the cosine function equation:
cos(x) = 0.5 * cos(3x) + 0
Therefore, the cosine function that has a maximum of 0.5, a minimum of -0.5, and a period of 2π/3 is:
cos(x) = 0.5 * cos(3x)
To determine the cosine function with specific criteria, we can use the general form of the cosine function:
f(x) = A * cos(Bx - C) + D
Where:
A = amplitude
B = frequency
C = phase shift
D = vertical shift
In this case, we know the following criteria:
- Maximum value of 0.5: This indicates the amplitude of the function, so A = 0.5.
- Minimum value of -0.5: This indicates the amplitude of the function, so A = 0.5.
- Period of 2π/3: This means that the function completes one cycle every 2π/3 units, or in other words, the frequency is 2π/3. Therefore, B = 2π/3.
To determine the phase shift (C) and vertical shift (D), we need more information. The given criteria do not provide this information.