Question

what is the measure of the angle labeled a in the scale drawing of the dog pen? a. sin⁻¹(9/15) b. cos⁻¹(9/15) c. tan⁻¹(9/15) d. tan⁻¹(15/9)

Explanation:

Step1: Recall trigonometric - inverse relationships

In a right - triangle, if the opposite side to angle \(A\) is \(a\), the adjacent side is \(b\), and the hypotenuse is \(c\), then \(\sin A=\frac{a}{c}\), \(\cos A = \frac{b}{c}\), and \(\tan A=\frac{a}{b}\). The inverse sine function \(\sin^{-1}(x)\) gives the angle whose sine is \(x\), \(\cos^{-1}(x)\) gives the angle whose cosine is \(x\), and \(\tan^{-1}(x)\) gives the angle whose tangent is \(x\).
If we assume that in the right - triangle related to the dog pen, the side opposite to angle \(A\) has length \(9\) and the hypotenuse has length \(15\).

Step2: Use the sine inverse formula

By the definition of the sine function \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, if \(\sin A=\frac{9}{15}\), then to find the measure of angle \(A\), we use the inverse - sine function. So \(A = \sin^{-1}(\frac{9}{15})\).

Answer:

A. \(\sin^{-1}(\frac{9}{15})\)