If 0≤x<2π, find all values of x that satisfy the following equation. cos(3x)=−√3/2

Question

Select the correct answer below:

x=4π3 and 5π3

x=5π6 and 7π6

x=5π18,7π18,19π18

x=4π9,5π9,10π9,11π9,16π9,17π9

x=5π18,7π18,17π18,19π18,29π18,31π18

Answer 1

Hi Heather,

to solve for x, you should first understand when cos(x) is negative - cos(x) is negative when the angle x is in the 2nd and 3rd quadrants.

value of cos(x) is sqrt3/2 for cos(30 degrees) or cos(pi/6)

Let’s consider an angle in the 2nd quadrant which has a reference angle of 30degrees - it would be

180-30=150degrees

or pi-pi/6= 5pi/6

adding 2n*pi to this we get 17pi/6, 29pi/6, 41pi/6 and so on

Since we are given cos(3x), we have to divide these values by 3, to get x.

so you will have 5pi/18, 17pi/18, 29pi/18, 41pi/18 and so on.

41pi/18 >2pi this can’t be a solution

Likewise if you take an angle in the 3rd quadrant which has a reference angle of pi/6, you have

pi+pi/6=7pi/6,

adding 2n*pi we get other angles 19pi/6,31pi/6, 43pi/6 and so on

dividing by 3, we have solutions for x as 7pi/18, 19pi/18, 31pi/18, 43pi/18, and so on

43pi/18 >2pi this can’t be a solution

hope this helps!