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:
- The leg opposite angle \(x\) is 3.
- 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 \(\cos y^\circ\):
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For \(\sin x^\circ\): \[ \sin x^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
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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.
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