Question

15
x = sin⁻

• 38.6

(right triangle with angle x°, hypotenuse 37)
(right triangle with legs 40, and hypotenuse 41, angle x° at the base)

Explanation:

Step1: Identify the triangle type

The bottom triangle is a right - triangle with hypotenuse \(c = 41\) and opposite side to angle \(x\) is \(a=40\). We can use the sine function which is defined as \(\sin(x)=\frac{\text{opposite}}{\text{hypotenuse}}\) in a right - triangle.

Step2: Apply the sine formula

We know that \(\sin(x)=\frac{40}{41}\). Then to find \(x\), we take the inverse sine (arcsin) of \(\frac{40}{41}\), so \(x = \sin^{- 1}(\frac{40}{41})\).

Step3: Calculate the value

Using a calculator, \(\sin^{-1}(\frac{40}{41})\approx\sin^{-1}(0.9756)\approx77.3^{\circ}\) (approximate value, the calculation may vary slightly depending on the calculator's precision). For the middle triangle, if we assume it's an isosceles right - triangle (since two sides are marked as equal and it's a right - triangle), the non - right angles would be \(45^{\circ}\), but from the given \(\sin^{-1}\) calculation in the top part, if we assume the opposite side is 15 and hypotenuse is, say, 24 (since \(\sin^{-1}(\frac{15}{24})=\sin^{-1}(0.625)\approx38.68^{\circ}\)), we use the same \(\sin(x)=\frac{\text{opposite}}{\text{hypotenuse}}\) formula.

Answer:

For the bottom right - triangle with hypotenuse 41 and opposite side 40, \(x\approx77.3^{\circ}\); for the middle triangle (isosceles right - triangle) \(x = 45^{\circ}\); for the top - related calculation with opposite side 15 and hypotenuse 24 (assumed), \(x\approx38.7^{\circ}\) (the specific answer depends on the exact triangle dimensions, but the method is using \(\sin(x)=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(x=\sin^{-1}(\frac{\text{opposite}}{\text{hypotenuse}})\))