Questions: Determine a general formula (or formulas) for the solution to the following equation. Then, determine the specific solutions (if any) on the interval [0,2 π). sin θ=0

Solution Steps

To solve the equation \(\sin \theta = 0\), we need to find the values of \(\theta\) where the sine function equals zero. The sine function is zero at integer multiples of \(\pi\). Therefore, the general solution is \(\theta = n\pi\), where \(n\) is an integer. To find the specific solutions in the interval \([0, 2\pi)\), we need to identify the integer values of \(n\) that satisfy this condition.

The equation \(\sin \theta = 0\) has a general solution given by: \[ \theta = n\pi \] where \(n\) is any integer.

To find the specific solutions within the interval \([0, 2\pi)\), we evaluate the general solution for integer values of \(n\):

• For \(n = 0\): \(\theta = 0 \cdot \pi = 0\)
• For \(n = 1\): \(\theta = 1 \cdot \pi = \pi\)
• For \(n = 2\): \(\theta = 2 \cdot \pi = 2\pi\) (not included in the interval)

Thus, the specific solutions in the interval \([0, 2\pi)\) are: \[ \theta = 0 \quad \text{and} \quad \theta = \pi \]

Final Answer