For a particular angle Θ, the cosine function f(x) = a cos b(Θ) has the following values within one cycle of the function:
f(0) = 4; f(pi/4) = 0; f(pi/2) = -4; f(3pi/4) = 0; f(pi) = 4
What is the rule for the cosine function?
A. y = 2 cos 4θ
B. y = 4 cos 2θ
C. y = –4 cos 3θ
D. y = one-fourth cos 2θ
B. y = 4 cos 2θ
To find the rule for the cosine function, we need to analyze the given values and determine the relationship between the angle Θ and the function values.
By observing the given values within one cycle of the function:
f(0) = 4
f(pi/4) = 0
f(pi/2) = -4
f(3pi/4) = 0
f(pi) = 4
We can notice that the function value transitions from positive (4) to zero, then to negative (-4), and back to zero, and finally back to positive (4) again.
This behavior suggests that the cosine function is being scaled by a factor of 4 and has a frequency of two complete cycles within the range of 0 to 2π (one cycle).
Therefore, the rule for the cosine function that matches this behavior is:
y = 4 cos 2θ
Hence, the correct option is B. y = 4 cos 2θ.