Find all solutions of the equation in the interval [0,2pi).

sqrt(3)cottheta-1= 0
Write your answer in radians in terms of pi.
If there is more than one solution, separate them with commas.

Does sqrt(3) mean the cube root?

Are you taking a root of cot theta, or (cotheta -1)? or 3 cot theta?

Hahan

To solve the equation sqrt(3)cot(theta) - 1 = 0, we can isolate cot(theta) by adding 1 to both sides of the equation:

sqrt(3)cot(theta) = 1

Next, divide both sides of the equation by sqrt(3):

cot(theta) = 1/sqrt(3)

Now, we can find the values of theta that satisfy this equation on the given interval [0, 2pi).

The cotangent function is equal to the reciprocal of the tangent function, so we can rewrite this equation as:

tan(theta) = sqrt(3)

From the unit circle, we know that the tangent function equals sqrt(3) at two angles in the interval [0, 2pi): pi/3 and 4pi/3.

Therefore, the solutions to the equation sqrt(3)cot(theta) - 1 = 0 in the interval [0, 2pi) are:

theta = pi/3, 4pi/3

Note that these values are already in radians and in terms of pi.

To find all solutions of the equation in the interval [0, 2π), we need to isolate the variable θ.

Given the equation: √3cotθ - 1 = 0

Let's solve it step by step:

1. Start by adding 1 to both sides of the equation to isolate the square root term:
√3cotθ = 1

2. Now, divide both sides by √3 to isolate the cotangent term:
cotθ = 1/√3

3. To find the values of cotθ, we need to find the corresponding values of θ. Recall that the cotangent function is the reciprocal of the tangent function. So, cotθ = 1/√3 is equivalent to tanθ = √3.

4. The value of √3 indicates that θ is in the first or third quadrant. Since the interval is [0, 2π), we only need to consider the first revolution of angles.

5. Using the unit circle or a calculator, we can determine that the angle θ whose tangent is equal to √3 is π/3 or 60 degrees.

6. However, we need to consider both the positive and negative values of cotθ, as cotθ is positive in the first quadrant and negative in the third quadrant. Therefore, θ can also be -π/3 or -60 degrees.

7. In radians, the solutions are θ = π/3 and θ = -π/3.

So, the solutions of the equation in the interval [0, 2π) are θ = π/3 and θ = -π/3 (or θ = 60 degrees and θ = -60 degrees in degrees).