Question

using sin:
s-o-h
(first right triangle: right angle at bottom left, angle 29° at bottom right, hypotenuse 34, vertical leg x)
(second right triangle: right angle at top right, angle 48° at top left, vertical leg 24, hypotenuse x)
(third right triangle: right angle at top right, hypotenuse 25, horizontal leg 7, angle x° at bottom right)

Explanation:

Step1: First Triangle (Solve for \( x \))

We use the sine function: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). For the first triangle, \( \theta = 29^\circ \), hypotenuse \( = 34 \), opposite side \( = x \). So \( \sin(29^\circ) = \frac{x}{34} \). Then \( x = 34 \times \sin(29^\circ) \). Calculating \( \sin(29^\circ) \approx 0.4848 \), so \( x \approx 34 \times 0.4848 \approx 16.48 \).

Step2: Second Triangle (Solve for \( x \))

For the second right triangle, angle \( = 48^\circ \), opposite side to the angle is \( 24 \), hypotenuse \( = x \). So \( \sin(48^\circ) = \frac{24}{x} \). Then \( x = \frac{24}{\sin(48^\circ)} \). \( \sin(48^\circ) \approx 0.7431 \), so \( x \approx \frac{24}{0.7431} \approx 32.3 \).

Step3: Third Triangle (Solve for \( x^\circ \))

For the third triangle, hypotenuse \( = 25 \), opposite side to angle \( x \) is \( 7 \). So \( \sin(x^\circ) = \frac{7}{25} = 0.28 \). Then \( x = \arcsin(0.28) \approx 16.3^\circ \).

Answer:

First triangle: \( x \approx 16.5 \) (rounded), Second triangle: \( x \approx 32.3 \), Third triangle: \( x \approx 16.3^\circ \) (answers may vary slightly due to rounding)