Question

Consider the solutions of the following equation over the interval 0 to 2π, or the interval 0° to 360°. Of the choices shown, which is not a solution to the equation? 3 cot² 0-1=0 O All of the cho

Answer 1

Step-by-step explanation:

We can simplify the given equation as follows:

3 cot² θ - 1 = 0

3 cot² θ = 1

cot² θ = 1/3

Taking the square root of both sides, we get:

cot θ = ±1/√3

Using the definition of cotangent, we know that:

cot θ = cos θ / sin θ

So we can rewrite the above equation as:

cos θ / sin θ = ±1/√3

Multiplying both sides by √3 and simplifying, we get:

cos θ = ±sin θ / √3

Squaring both sides and using the identity sin² θ + cos² θ = 1, we get:

1/3 = sin² θ + (sin θ / √3)²

Multiplying both sides by 3, we get:

1 = 3 sin² θ + sin² θ

4 sin² θ = 1

sin θ = ±1/2

Therefore, the possible solutions for θ are:

θ = 30°, 150°, 210°, 330°

Now we can check the given choices to see which one is not a solution to the equation:

- 45°: not a solution, since sin 45° = √2/2 ≠ ±1/2

- 150°: a solution, since sin 150° = -1/2 and cos 150° = -√3/2

- 210°: a solution, since sin 210° = -1/2 and cos 210° = √3/2

- 330°: a solution, since sin 330° = 1/2 and cos 330° = -√3/2

Therefore, the choice that is not a solution to the equation is -45°.