3 cos(x) = 3/2

Question

_{Refer to the graph of y = cos(x) to find the exact values of x in the interval [0, 4π] that satisfy the equation. Enter solutions in a list separated by commas. Exact answers only: decimal points appearing in your answer will be marked incorrect.)}

 

3 cos(x) = 3/2

 

x= 

Answer 1

Multiply both sides of the equation by 1/3.

 

cos(x) = 1/2

 

 

Take the inverse of cosine.

 

x₁ = cos^-1(1/2)

 

 

Since cosine is positive in the 1st and 4th quadrants, your other angle will be

 

x₂ = 2π - cos^-1(1/2)

 

 

Because we are going round the circle twice.  We have 2 more values of x.

 

x₃ = 2π + cos^-1(1/2)   ----> 1st quadrant 2nd lap

 

x₄ = 4π - cos^-1(1/2)    ---->  4th quadrant 2nd lap

Answer 2

3cosx = 3/2

 

cosx = 1/2

 

cosx is positive in quadrant 1 and 4

 

Solution between 0 and 2π in quadrant 1:  π/3

 

Solution between 0 and 2π in quadrant 4:  2π - π/3 = 5π/3

 

Add 2π to each of the solutions above to get the solutions between 2π and 4π.  They are 7π/3 and 11π/3.

 

Solutions:  π/3, 5π/3, 7π/3, 11π/3