Question

in which triangle is the measure of the unknown angle, x, equal to the value of sin^{-1}(\frac{5}{8.3})?

Explanation:

Step1: Recall the sine function definition

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\theta = x\), we have \(\sin x=\frac{5}{8.3}\), so \(x = \sin^{-1}(\frac{5}{8.3})\). This means the side opposite to angle \(x\) has length \(5\) and the hypotenuse has length \(8.3\).

Step2: Analyze each triangle

• In the first triangle: The side opposite to \(x\) is \(5\) and the hypotenuse is \(8.3\). Using the formula \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{5}{8.3}\), then \(x=\sin^{-1}(\frac{5}{8.3})\).
• In the second triangle: Using \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), the side opposite to \(x\) is \(5\) but the hypotenuse is not \(8.3\) (the side of length \(8.3\) is not the hypotenuse).
• In the third triangle: Using \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), the side opposite to \(x\) is not \(5\) (the side of length \(5\) is not opposite to \(x\)).
• In the fourth triangle: Using \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), the side opposite to \(x\) is not \(5\) (the side of length \(5\) is adjacent to \(x\) in the right - triangle context for the sine ratio).

Answer:

The first triangle.