Question

find all solutions to the equation.

\\( \sin \theta - 1 = - 1 \\)

write your answer in radians in terms of \\( \pi \\), and use the \or\ button as necessary.

example: \\( \theta = \frac { \pi } { 5 } + 2 k \pi , k \in \mathbb { z } \\) or \\( \theta = \frac { \pi } { 7 } + k \pi , k \in \mathbb { z } \\)

Explanation:

Step1: Solve for $\sin\theta$

Add \(1\) to both sides of the equation \(\sin\theta - 1=-1\).
$$\sin\theta-1 + 1=-1 + 1$$
$$\sin\theta=0$$

Step2: Find the general solution for \(\theta\)

We know that \(\sin\theta = 0\) when \(\theta = k\pi\), where \(k\in\mathbb{Z}\) (because the sine function has a value of \(0\) at \(0,\pi,2\pi,\cdots\) and \(-\pi,-2\pi,\cdots\)).

Answer:

\(\theta = k\pi,k\in\mathbb{Z}\)