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 definition of sine function

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

Step2: Analyze each triangle

• For the first triangle: The side opposite to \(x\) is \(5\) and the hypotenuse is \(8.3\).
• For the second triangle: Using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), if we consider the right - angle, the side opposite to \(x\) is \(5\) but the hypotenuse is not \(8.3\) (the side with length \(8.3\) is adjacent to \(x\)).
• For the third triangle: The side with length \(5\) is adjacent to \(x\) (not opposite).
• For the fourth triangle: The side with length \(5\) is adjacent to \(x\) (not opposite).

Answer:

The first triangle.