Question
Find all angles, \( 0^{\circ} \leq \theta<360^{\circ} \), that solve the following equation. \[ \sin \theta=-1 \]
Ask by Potter Campos. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The angle that solves \( \sin \theta = -1 \) within \( 0^{\circ} \leq \theta < 360^{\circ} \) is \( \theta = 270^{\circ} \).
Solution
1. The sine function reaches the value \(-1\) at the angle where the terminal point on the unit circle is \((0, -1)\).
2. In one full cycle, \(0^{\circ} \leq \theta < 360^{\circ}\), this occurs at
\[
\theta = 270^{\circ}.
\]
3. Therefore, the solution to the equation
\[
\sin \theta = -1
\]
within the interval is
\[
\theta = 270^{\circ}.
\]
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Simplify this solution Beyond the Answer
To find all angles \( \theta \) that satisfy the equation \( \sin \theta = -1 \) within the interval \( 0^{\circ} \leq \theta < 360^{\circ} \), we need to consider where the sine function attains its minimum value of -1. The sine function reaches -1 at \( \theta = 270^{\circ} \). Thus, the only solution in the specified interval is: \[ \theta = 270^{\circ} \] This is the point on the unit circle where the terminal side of the angle is directly downward along the y-axis.
