Answer:
Hope this helps ^^
Explanation:
Part A:
To determine the values where the pogo stick's spring will be equal to its non-compressed length, we need to set the function f(θ) equal to zero since the non-compressed length is represented by zero in this case.
Given:
f(θ) = 2cos(θ) + √3
To find the values where f(θ) = 0, we solve the equation:
2cos(θ) + √3 = 0
Subtracting √3 from both sides:
2cos(θ) = -√3
Dividing both sides by 2:
cos(θ) = -√3/2
Using the unit circle or trigonometric values, we find that θ can take on the values of π/6 and 11π/6.
Therefore, the values where the pogo stick's spring will be equal to its non-compressed length are θ = π/6 and θ = 11π/6.
Part B:
If we double the angle, θ becomes 2θ. So the function becomes:
f(2θ) = 2cos(2θ) + √3
In the interval [0, 2π), we need to find the solutions for f(2θ) = 0.
Simplifying the equation:
2cos(2θ) + √3 = 0
Dividing the equation by 2:
cos(2θ) = -√3/2
Using the double-angle identity for cosine, we have:
cos(2θ) = 2cos^2(θ) - 1
Replacing cos(2θ) in the equation:
2cos^2(θ) - 1 = -√3/2
Multiplying the equation by 2:
4cos^2(θ) - 2 = -√3
Adding √3/2 to both sides:
4cos^2(θ) = -√3/2 + 1
Simplifying further:
cos^2(θ) = (-√3 + 2)/4
Taking the square root of both sides:
cos(θ) = ±√((-√3 + 2)/4)
Using the unit circle or trigonometric values, we can find the corresponding angles in the interval [0, 2π). These angles will be the solutions for f(2θ) = 0.
Comparing to the original function, the solutions for f(2θ) = 0 will have different values compared to the solutions for f(θ) = 0. The angles will be different due to the doubling of the angle, resulting in a different function.