Question

PLEASE HELP!

Let f be the function given by f (x) = (create an original sinusoidal function with an amplitude not equal to 1, a period not equal to 2π, and non-zero phase and vertical shifts).

ex: F of x equals negative one half times sine of quantity 3 times x plus pi over 2 end quantity minus 2


Part A: State the amplitude and vertical shift.


Part B: Determine the period of f (x), showing all necessary calculations.


Part C: Calculate the phase shift of the sinusoidal function with proper mathematical justification.


Part D: Graph the sinusoidal function by hand, using your answers from parts A–C.


Choose an angle θ, in radians, such that 2π < θ < 4π . Let θ = (create an original angle measure).


ex: Theta equals 13 pi over 6


Part E: Determine the exact value of cos θ using the sum formula. Show all necessary mathematical work.


Part F: Determine the exact value of sin θ using the difference formula. Show all necessary mathematical work.


Part G: Calculate the exact value of tan 2θ, using your answers from parts E – F.

Answer 1

The equation of the function f(x) is f(x) = 2 sin(π/2(x + 6)) - 3

How to create the sine function?

A sine function is represented as:

f(x) = A sin(B(x + C)) + D

Where

A = Amplitude

Period = 2π/B

C = Phase shift

D = Vertical shift

The requirements in the question are:

• Amplitude not equal to 1
• Period not equal to 2π
• Non-zero phase and vertical shifts

So, we can use the following assumptions

A = 2

Period = 4

C = 6

D = -3

So, we have:

f(x) = 2 sin(B(x + 6)) - 3

The value of B is

4 = 2π/B

This gives

B = π/2

So, we have:

f(x) = 2 sin(π/2(x + 6)) - 3

The amplitude, vertical shift, period of f(x)and the phase shift

Using the representations in (a), we have:

• Amplitude = 2
• Vertical shift = -3
• Period = 4
• Phase shift = 6

The graph of the function

See attachment for the graph of f(x)

The value of cos θ

Let θ = 3π

So, we have:

cos(3π)

This is calculated as:

cos(3π) = cos(2π + π)

Expand

cos(3π) = cos(2π) *cos(π) - sin(2π) *sin(π)

Evaluate

cos(3π) = -1

The value of sin θ

Let θ = 3π

So, we have:

sin(3π)

This is calculated as:

sin(3π) = sin(2π + π)

Expand

sin(3π) = sin(2π) *cos(π) + cos(2π) *sin(π)

Evaluate

sin(3π) = 0

The value of tan 2θ

Let θ = 3π

So, we have:

tan(2 * 3π)

tan(6π)

This is calculated as:

tan(6π) = tan(3π + 3π)

Evaluate

tan(3π) = 0

Read more about sinusoidal functions at:

https://brainly.com/question/10700288

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