Use inverse functions where needed to find all solutions of the equation in the interval [0,2π).
A)
B)
C)
D)

Question

Use inverse functions where needed to find all solutions of the equation in the interval [0,2π). ​  ​
A)​ 
B)​ 
C)​ 
D)​

Quizplus · Accepted Answer

We start by using the inverse function of cosine:

cos^{-1}(cos(x))=x

cos^{-1}(\frac{1}{2})=\frac{\pi}{3} and cos^{-1}(-\frac{1}{2})=\frac{2\pi}{3}

Now we have:

cos(x)=\frac{1}{2}

x=cos^{-1}(\frac{1}{2})+2\pi k_1 and x=-cos^{-1}(\frac{1}{2})+2\pi k_2

where k_1 and k_2 are integers.

Similarly:

cos(x)=-\frac{\sqrt{3}}{2}

x=cos^{-1}(-\frac{\sqrt{3}}{2})+2\pi m_1 and x=-cos^{-1}(-\frac{\sqrt{3}}{2})+2\pi m_2

where m_1 and m_2 are integers.

In the interval [0,2π), the solutions are:

x=\frac{\pi}{3},\frac{5\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3}

Therefore, the only correct choice is B.

Use Inverse Functions Where Needed to Find All Solutions of the Equation