Question
For which values of $A$ and $B$ is $\sin A = \cot B$?
The correct answer is
$A = 0,\ B = \frac{\pi}{2}$
Equation Verification for $\sin A = \cot B$
The problem requires finding the values of $A$ and $B$ that satisfy the trigonometric equation $\sin A = \cot B$. We will test the given options.
Checking Specific Values
Let's evaluate the equation $\sin A = \cot B$ for the values provided in the options.
- Option 1: $A = 0, B = 0$
- $\sin A = \sin 0 = 0$
- $\cot B = \cot 0$, which is undefined.
- Therefore, this option is incorrect.
- Option 2: $A = \frac{\pi}{2}, B = \frac{\pi}{2}$
- $\sin A = \sin \frac{\pi}{2} = 1$
- $\cot B = \cot \frac{\pi}{2} = 0$
- Since $1 \neq 0$, this option is incorrect.
- Option 3: $A = 0, B = \frac{\pi}{2}$
- $\sin A = \sin 0 = 0$
- $\cot B = \cot \frac{\pi}{2} = 0$
- Since $0 = 0$, this option satisfies the equation.
- Option 4: $A = \frac{\pi}{2}, B = 0$
- $\sin A = \sin \frac{\pi}{2} = 1$
- $\cot B = \cot 0$, which is undefined.
- Therefore, this option is incorrect.
Conclusion on Values
The only values that satisfy the equation $\sin A = \cot B$ among the choices are $A = 0$ and $B = \frac{\pi}{2}$.
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