To find the value of angle \( x \) in the right triangle where the sides are given, we can apply the sine, cosine, or tangent functions.
Let’s apply the sine function since we have the length of the opposite side (across from \( x \)) and the hypotenuse.
In this right triangle:
- The length of the side opposite angle \( x \) is \( 21 \) (hypotenuse).
- The adjacent side (side next to angle \( x \)) is \( 15 \).
Using the sine function: \[ \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \sin(x) = \frac{21}{\text{hypotenuse}} \quad (\text{Hypotenuse } = 21) \]
To find the hypotenuse using the Pythagorean Theorem: \[ \text{Hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \] \[ \text{Hypotenuse}^2 = 21^2 + 15^2 \] \[ = 441 + 225 = 666 \] Taking the square root: \[ \text{Hypotenuse} = \sqrt{666} \approx 25.8 \]
Now, we can find angle \( x \) using the cosine relationship since we have the adjacent side: \[ \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{15}{\sqrt{666}} \] Calculate this ratio: \[ = \frac{15}{25.8} \approx 0.581 \]
Now, we can find angle \( x \): \[ x = \cos^{-1}(0.581) \approx 54.4^\circ \]
However, using sine to confirm: \[ x = \sin^{-1}\left(\frac{21}{\sqrt{666}}\right) \approx 36.1^\circ \]
Rounding accordingly, we find \( x \approx 36^\circ \).
The final answer is: \[ \boxed{36^\circ} \]
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