Question

Study the work shown to find the exact solutions of this trigonometric equation:

2cos(x)−√3​=0

for 0 ≤ x ≤ 2π.

2cos(x)−√3​=0.
2cos(x)=√3​.
cos(x)=√3/2​​.
cos⁻¹(√3/2​​)=x.

Select all that are solutions of the equation over the specified domain.

a. x=π/6​
b. x=5π6​
c. x=7π/6​
d. x=11π6​.

Answer 1

The exact solutions of the equation 2cos(x) - √3 = 0 within the specified domain 0 ≤ x ≤ 2π are x = π/6 and x = 11π/6. These angles correspond to 30° and 330° on the unit circle.

Step-by-step explanation:

To solve the trigonometric equation 2cos(x) - √3 = 0, we begin by isolating cos(x) to obtain cos(x) = √3/2. The cosine inverse function, denoted as cos⁻¹, is used to find the angles at which the cosine value is √3/2 in the specified domain of 0 ≤ x ≤ 2π.

The cosine of an angle yields √3/2 at two places in the unit circle within one full rotation: x = π/6 and x = 11π/6, which correspond to angles 30° and 330° when measured in degrees.

The other options provided, x = 5π/6, x = 7π/6, do not yield a cosine value of √3/2, so they are not solutions to the equation within the given domain.

Therefore, the correct solutions are:

• x = π/6
• x = 11π/6