Question

Use the image below to answer the following question. Find the value of sin x° and cos y°. What relationship do the ratios of sin x° and cos y° share? (10 points)

A right triangle is shown with one leg measuring 4 and another leg measuring 3. The angle across from the leg measuring 3 is marked x degrees, and the angle across from the leg measuring 4 is marked y degrees.

Answer 1

To determine the values of \(\sin x^\circ\) and \(\cos y^\circ\) based on the right triangle you've described, we can use the definitions of sine and cosine.

In the right triangle:

1. The leg opposite angle \(x\) is 3.
2. The leg opposite angle \(y\) is 4.

Since this is a right triangle, we can find the length of the hypotenuse \(h\) using the Pythagorean theorem:

\[
h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Now we can find the values of \(\sin x^\circ\) and \(\cos y^\circ\):

1. **For \(\sin x^\circ\)**:
\[
\sin x^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}
\]

2. **For \(\cos y^\circ\)**:
\[
\cos y^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}
\]

Now, let's examine the relationship between \(\sin x^\circ\) and \(\cos y^\circ\). Since \(x\) and \(y\) are complementary angles in a right triangle, we have:

\[
y = 90^\circ - x
\]

Therefore, from the co-function identity:

\[
\cos y^\circ = \sin(90^\circ - y) = \sin x^\circ
\]

This shows that:

\[
\sin x^\circ = \cos y^\circ
\]

In summary:
- \(\sin x^\circ = \frac{3}{5}\)
- \(\cos y^\circ = \frac{4}{5}\)

And they share the relationship that \(\sin x^\circ = \cos y^\circ\), consistent with the complementary angles in a right triangle.