Question

Solve 5cos(5x) = 2 for the smallest three positive solutions.

Answer 1

Given the equation:

$5cos(5x)=2$

Let's solve the equation for the smallest three positive solutions.

The first step is to divide both sides of the equation by 5:

$\begin{gathered} \frac{5cos(5x)}{5}=\frac{2}{5} \\ \\ cos(5x)=0.4 \end{gathered}$

Take the cos inverse of both sides:

$5x=cos^{-1}(0.4)+2\pi n$

Where n is any integer.

Now, divide both sides by 5:

$\begin{gathered} \frac{5x}{5}=\frac{cos^{-1}(0.4)}{5}+\frac{2\pi n}{5} \\ \\ x_1=\frac{cos^{-1}(0.4)}{5}+\frac{2\pi n}{5} \\ \\ x_2=\frac{2\pi}{5}-\frac{cos^{-1}(0.4)}{5}+\frac{2\pi n}{5} \\ \\ When\text{ n = 0:} \\ x_1=\frac{cos^{-1}(0.4)}{5}+\frac{2\pi(0)}{5}=0.23 \\ \\ x_2=\frac{2\pi}{5}-\frac{cos^{-1}(0.4)}{5}+\frac{2\pi(0)}{5}=1.02 \end{gathered}$

When n = 1:

$\begin{gathered} x_1=\frac{cos^{-1}(0.4)}{5}+\frac{2\pi(1)}{5}=1.49 \\ \\ x_2=\frac{2\pi}{5}-\frac{cos^{-1}(0.4)}{5}+\frac{2\pi(1)}{5}=-0.231 \end{gathered}$

When n = 2:

$\begin{gathered} x_1=\frac{cos^{-1}(0.4)}{5}+\frac{2\pi(2)}{5}=2.75 \\ \\ x_2=\frac{2\pi}{5}-\frac{cos^{-1}(0.4)}{5}+\frac{2\pi(2)}{5}=1.024 \end{gathered}$

Therefore, the smallest three positive solutions are:

x = 0.23, 1.02, 1.49

ANSWER:

0.23, 1.02, 1.49