Question
Solve the equation on the interval \( 0 \leq \theta<2 \pi \). \[ \begin{array}{l}\sin \theta+3=5 \\ \text { What are the solutions in the interval } 0 \leq \theta<2 \pi \text { ? Select the correct choice and fill in any answer boxes in your choice below. } \\ \text { A. The solution set is }\{\text {. } \\ \text { (Simplify your answer. Type an exact answer, using } \pi \text { as needed. Type your answer in radians. Use integers or fraction } \\ \text { to separate answers as needed.) } \\ \text { B. There is no solution. }\end{array} \text {. } \]
Ask by Chen Love. in the United States
Dec 03,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
B. There is no solution.
Solution
To solve the equation
\[
\sin \theta + 3 = 5,
\]
we first isolate \(\sin \theta\):
\[
\sin \theta = 5 - 3,
\]
\[
\sin \theta = 2.
\]
Next, we need to determine if there are any values of \(\theta\) for which \(\sin \theta = 2\). The sine function has a range of \([-1, 1]\), meaning it can only take values between -1 and 1. Since 2 is outside this range, there are no angles \(\theta\) for which \(\sin \theta = 2\).
Thus, the conclusion is:
\[
\text{B. There is no solution.}
\]
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Simplify this solution Mind Expander
To solve the equation \(\sin \theta + 3 = 5\), we first isolate the sine function by subtracting 3 from both sides: \[ \sin \theta = 5 - 3 \] \[ \sin \theta = 2 \] However, the sine function has a range of \([-1, 1]\). Since 2 is outside this range, there are no values of \(\theta\) for which \(\sin \theta = 2\). So, the solution to this equation is: \[ \text{B. There is no solution.} \]
