Question

find all solutions of cosx-\sqrt(1-3cos^(2)x)=0 a. 60° + n360°, 300° + n360° b. 30° + n360°, 210° + n360° c. 30° + n360°, 330° + n360° d. 60° + n360°, 120° + n360°

Answer 1

Greetings from Brasil...

The equation is:

COS X - √(1 - 3.COS² X) = 0

putting COS X for the 2nd member

- √(1 - 3.COS² X) = - COS X ×(- 1)

√(1 - 3.COS² X) = COS X everything squared

[√(1 - 3.COS² X)]² = (COS X)²

1 - 3.COS² X = COS² X

1 - 3.COS² X - COS² X = 0

1 - 4.COS² X = 0

making Y = COS X

1 - 4Y² = 0

- 4Y² = - 1 ×(- 1)

4Y² = 1

Y² = 1/4

Y = ±√1/4

Y = ± 1/2

so, as we saw above

Y = COS X

1/2 = COS X and - 1/2 = COS X

so to get COS X = ± 1/2, then

X = π/3 = 60 (cos +)

X = 2π/3 = 120 (cos -)

X = 4π/3 = 240 (cos -)

X = 5π/3 = 300 (cos +)

So the only option that includes + and - its:

60° + n360° , 120° + n360°