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

When n = 1:

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

When n = 2:

`\begin{gathered} x_1=(cos^(-1)(0.4))/(5)+(2\pi(2))/(5)=2.75 \\ \\ x_2=(2\pi)/(5)-(cos^(-1)(0.4))/(5)+(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