Question
Q. If \( \sin x=\cos y \), what must be true about \( x \) and \( y \) ? answer choices \( x+y=90^{\circ} \) \( x=y \) \( x=2 y \) \( x+y=180^{\circ} \)
Ask by Bowen Mitchell. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x + y = 90^\circ \)
Solution
To solve the problem, we start with the given equation:
\[
\sin x = \cos y
\]
We can use the co-function identity, which states that:
\[
\sin x = \cos(90^\circ - x)
\]
This means that if \(\sin x = \cos y\), then we can set:
\[
y = 90^\circ - x
\]
Now, if we rearrange this equation, we get:
\[
x + y = 90^\circ
\]
Thus, the correct answer is:
\[
\boxed{x + y = 90^\circ}
\]
Now, let's analyze the other options to confirm that they are not true:
1. **\( x = y \)**: This is not necessarily true since \( \sin x \) and \( \cos y \) can be equal without \( x \) and \( y \) being the same angle.
2. **\( x = 2y \)**: This does not follow from the given equation and is not a general truth.
3. **\( x + y = 180^\circ \)**: This is also not true based on the co-function identity we used.
Therefore, the only statement that must be true about \( x \) and \( y \) is:
\[
x + y = 90^\circ
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
When \( \sin x = \cos y \), this indicates a special relationship between the angles in a right triangle context. In fact, the sine of an angle is equal to the cosine of its complement! Therefore, for \( x + y = 90^{\circ} \) to hold true, it means that \( x \) and \( y \) are complementary angles, leading us to the conclusion that this is the correct answer. Engaging this concept further, you can visualize it with a right triangle where one angle is \( x \) and the other is \( y \). The angle \( y \) is simply \( 90 - x \), hence \( \sin x \) equals the cosine of \( 90 - x \) or \( \cos y \). It's a fun little dance of angles in the realm of trigonometry!
