Question

Find the solutions in the interval [0, 2π) for the equation cscθ - cotθ = sinθ. Enter your answers as a comma-separated list.

Answer 1

To solve the equation cscθ - cotθ = sinθ, trigonometric identities are used to simplify it, resulting in the solutions π/2 and 3π/2, which are tested back into the equation to ensure they are reasonable.

Step-by-step explanation:

To find the solutions to the equation cscθ - cotθ = sinθ in the interval [0, 2π), we must first identify the known trigonometric identities that can be useful for simplifying the equation:

• cscθ is the reciprocal of sinθ, so cscθ = 1/sinθ.
• cotθ is the reciprocal of tanθ, and therefore the quotient of cosθ over sinθ, so cotθ = cosθ/sinθ.

Let us substitute these into the given equation to simplify it:

1. 1/sinθ - (cosθ/sinθ) = sinθ
2. Multiply through by sinθ to get rid of the denominators: 1 - cosθ = sin²θ
3. Now, we know that sin²θ = 1 - cos²θ from the Pythagorean identity.
4. Substituting this into our equation, we get 1 - cosθ = 1 - cos²θ, which simplifies to cosθ = cos²θ.
5. Setting cosθ equal to zero, we find the angles where cosθ is zero within the given interval.

To check if the answer is reasonable, we set θ to those found angles and see if they satisfy the original equation. The angles where cosθ is zero within [0, 2π) are π/2 and 3π/2.

Upon testing these solutions back into the original equation, we can confirm that they make sense and therefore are the correct solutions.