Solve 4 cos (5x)=3 for the smallest three positive solutions.

Question

How would I solve this? If positive, please explain how you found your answer.

Answer 1

Let θ = 5x

Then the problem is:

4cos(θ) = 3

cos(θ) = 0.75

θ = cos^-1(0.75)

Using your calculator θ = 0.7227

Drawing this out on the unit circle:

You can see that the same x-value is achieved with θ = -0.7227 radians as well. And the same value of cosine would be achieved by:

0.7227 + 2π (which is θ = 7.0059)

-0.7227 + 2π (which is θ = 5.5605)

0.7227 + 4π (which is θ = 13.2891)

-0.7227 + 4π (which is θ = 11.8436)

etc

Now, make the conversion back to the "x domain":

Since θ = 5x then:

5x = 0.7227 or x = 0.145

5x = 7.0059 or x = 1.401

5x = 5.5605 or x = 1.112

5x = 13.2891 or x = 2.658

5x = 11.8436 or x = 2.369

etc

Since you are being asked for the three smallest positive solutions, they would by:

x = 0.145

x = 1.112

x = 1.401