Calc 1 Trig Equation

Question

Find all values of x in the interval [0, 2π] that satisfy the following equation.
24 cot²(x) = 8

 

I think the first step is to solve for cot(x) which is equal to \sqrt{3} / 3. I don't know what to do from here. Thanks for any help you can give. 

Answer 1

cot²x = 1/3   So, tan²x = 3

 

tanx = ±√3

 

If tanx = √3, then x lies in quadrant 1 or 3

 

    x = Tan^-1(√3) = 60° or x = 180° + 60° = 240°

 

If tanx = -√3, then x lies either in quadrant 2 or quadrant 4

 

    x= 180° - 60° = 120° or x = 360° - 60° = 300°

 

Solutions between 0° and 360°:  60°, 120°, 240°, 300°

 

Answer 2

24cot²(x)=8

cot²(x)=8/24=1/3

cot(x)=1/√3

1/√3=√3/3

cot(x)=√3/3

cot=cos/sin

cot is positive in the 1st and 3rd quadrants

look on the unit circle...

at ∏/3=60º, cos=1/2 and sin=√3/2

cot=cos/sin=(1/2)/(√3/2)=(1/2)(2/√3)=1/√3=√3/3

therefore x=60º

now from 60º, go 180º to 240º (4∏/3)

at 240º, cos=(-1/2) and sin=(-√3/2)

therefore cos/sin, which again is the cot, equals √3/3 again because a negative divided by a negative is a positive

therefore the two solutions are x=60º and x=240º