Question

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find the value of x.
24.7
x°
10.3
degrees

Explanation:

Step1: Identify trigonometric ratio

In a right triangle, we can use the cosine function. The adjacent side to angle \( x \) is \( 10.3 \), and the hypotenuse is \( 24.7 \). So, \( \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{10.3}{24.7} \).

Step2: Calculate the angle

First, compute \( \frac{10.3}{24.7} \approx 0.417 \). Then, take the arccosine (inverse cosine) of \( 0.417 \) to find \( x \). Using a calculator, \( x = \cos^{-1}(0.417) \approx 65.3^\circ \) (or check if sine was intended: opposite is 10.3? Wait, no, wait: wait, the right angle, so the sides: hypotenuse is 24.7, one leg is 10.3. Wait, maybe I mixed adjacent and opposite. Wait, angle \( x \): the leg adjacent to \( x \) is the vertical leg (10.3) or horizontal? Wait, the right angle is at the bottom right, so the triangle has right angle, vertical leg 10.3, hypotenuse 24.7, so the angle \( x \) is at the top, so the adjacent side to \( x \) is the vertical leg (10.3), hypotenuse 24.7. So \( \cos(x) = \frac{10.3}{24.7} \). Alternatively, if the horizontal leg is, say, let's recalculate. Wait, maybe I made a mistake. Wait, let's check: in a right triangle, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). So angle \( x \): the sides: hypotenuse is 24.7, the side adjacent to \( x \) is 10.3 (the vertical leg), so \( \cos(x) = 10.3 / 24.7 \approx 0.417 \). Then \( x = \arccos(0.417) \approx 65.3^\circ \). Wait, but maybe it's sine? Wait, no, the opposite side to \( x \) would be the horizontal leg. Wait, maybe I mixed up. Wait, let's draw the triangle: right angle at the bottom right, so vertices: bottom right (right angle), top (angle \( x \)), bottom left. So the sides: bottom left to top is hypotenuse (24.7), bottom right to top is vertical leg (10.3), bottom left to bottom right is horizontal leg. So angle \( x \) is at the top, between hypotenuse and vertical leg. So adjacent to \( x \) is vertical leg (10.3), hypotenuse (24.7). So \( \cos(x) = 10.3 / 24.7 \). So calculating that: 10.3 divided by 24.7 is approximately 0.417. Then arccos(0.417) is approximately 65.3 degrees. Alternatively, if we use sine: opposite over hypotenuse. Opposite to \( x \) is the horizontal leg. Wait, but we don't know the horizontal leg. Wait, maybe the vertical leg is 10.3, hypotenuse 24.7, so the angle \( x \): let's check with sine. Wait, no, maybe I had adjacent and opposite reversed. Wait, angle \( x \): the side opposite to \( x \) is the horizontal leg, adjacent is vertical leg. So \( \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \), but we don't know opposite. Wait, no, we can use Pythagoras to find the horizontal leg: \( \text{horizontal leg} = \sqrt{24.7^2 - 10.3^2} \approx \sqrt{610.09 - 106.09} = \sqrt{504} \approx 22.45 \). Then \( \sin(x) = 22.45 / 24.7 \approx 0.909 \), so \( x = \arcsin(0.909) \approx 65.3^\circ \). Oh, same result! Because \( \sin(x) = \cos(90 - x) \), so both ways, we get approximately 65 degrees. Wait, but maybe the initial calculation was correct. So using a calculator, \( \arccos(10.3/24.7) \approx 65.3^\circ \).

Answer:

\( x \approx 65.3 \) degrees (or depending on calculator precision, maybe 65.2 or 65.4, but approximately 65.3)