Math Problem Statement

For a cosine function with amplitude A=0.75 and period T=10 , what is y(4) ?

Solution

We are given a cosine function with amplitude A=0.75A = 0.75 and period T=10T = 10. To find y(4)y(4), we can use the general form of the cosine function:

y(t)=Acos(2πT(tϕ))y(t) = A \cos\left( \frac{2\pi}{T}(t - \phi) \right)

where:

  • AA is the amplitude,
  • TT is the period,
  • tt is the time (or input variable),
  • ϕ\phi is the phase shift (which we assume is 0 unless specified otherwise).

Step 1: Determine the parameters

  • The amplitude A=0.75A = 0.75,
  • The period T=10T = 10.

Substitute these into the cosine function formula:

y(t)=0.75cos(2π10(t0))y(t) = 0.75 \cos\left( \frac{2\pi}{10}(t - 0) \right) Simplifying further:

y(t)=0.75cos(π5t)y(t) = 0.75 \cos\left( \frac{\pi}{5} t \right)

Step 2: Find y(4)y(4)

Now, substitute t=4t = 4 into the equation:

y(4)=0.75cos(π5×4)y(4) = 0.75 \cos\left( \frac{\pi}{5} \times 4 \right)

y(4)=0.75cos(4π5)y(4) = 0.75 \cos\left( \frac{4\pi}{5} \right)

Using a calculator to find the value of cos(4π5)\cos\left( \frac{4\pi}{5} \right):

cos(4π5)0.8090\cos\left( \frac{4\pi}{5} \right) \approx -0.8090

Thus:

y(4)=0.75×(0.8090)0.6068y(4) = 0.75 \times (-0.8090) \approx -0.6068

Final Answer:

y(4)0.607y(4) \approx -0.607

Would you like any further details or clarifications on this process? Here are some related questions that might interest you:

  1. How do you calculate the phase shift in a cosine function?
  2. What happens to the graph of a cosine function if the amplitude is negative?
  3. How do changes in the period affect the graph of a trigonometric function?
  4. Can you solve for the maximum and minimum values of the function for any given time?
  5. How would you adjust the function if the phase shift was nonzero?

Tip: When working with periodic functions like sine and cosine, it's helpful to understand how transformations such as amplitude, period, and phase shift affect the graph visually.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Functions
Cosine Function
Amplitude
Period

Formulas

y(t) = A * cos((2π/T) * (t - φ))
Cosine value evaluation

Theorems

Properties of Cosine Function

Suitable Grade Level

Grades 9-12