Question

For a particular angle theta, the cosine function f(x) = a cos b(theta) 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 4 theta

B) y= 4 cos 2 theta

C) y= -4 cos 3 theta

D) y= 1/4cos 2 theta

Answer 1

For function $f(x)=a\cos b\theta$ we have that $f(0) =4,\ f( \frac{\pi}{2} )=-4,\ f(\pi)=4$. This means that period of cosine function is π. Then from the formula for period $T= \frac{2\pi}{b}$ you obtain that $\pi= \frac{2\pi}{b}$ and b=2. 

So, the function is $f(x)=a\cos 2\theta$. If f(0)=4, then $a\cos0=4$ and, respectively, a=4.
Answer: $f(x)=4\cos 2\theta$.