Question

use the following diagram to answer question 3.

1. solve the above right triangle for the labelled unknowns. round your answers to one decimal place if necessary.

step 1: solve for x.

hint: \\( \theta = \\)
\\( \text { opposite } = \\)
\\( \text { adjacent } = \\)
\\( \text { trig ratio used: } \\)

Explanation:

Step1: Identify the trigonometric ratio for \(x\)

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 36^{\circ}\), the hypotenuse \(c = 65\) mm, and the adjacent side to \(\theta\) is \(x\).

Step2: Solve for \(x\)

Using the formula \(\cos\theta=\frac{x}{65}\), we can rewrite it as \(x = 65\times\cos(36^{\circ})\).
Since \(\cos(36^{\circ})\approx0.8090\), then \(x = 65\times0.8090=52.585\approx52.6\) mm.

Step3: Solve for \(y\)

Using the formula \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 36^{\circ}\), the hypotenuse \(c = 65\) mm, and the opposite side to \(\theta\) is \(y\).
Using the formula \(\sin\theta=\frac{y}{65}\), we can rewrite it as \(y = 65\times\sin(36^{\circ})\).
Since \(\sin(36^{\circ})\approx0.5878\), then \(y = 65\times0.5878 = 38.207\approx38.2\) mm.

Step4: Solve for angle \(A\)

In a right - triangle, the sum of angles is \(180^{\circ}\), and one angle is \(90^{\circ}\). So \(A+36^{\circ}+90^{\circ}=180^{\circ}\).
Then \(A=180^{\circ}-(90^{\circ} + 36^{\circ})=54^{\circ}\).

Answer:

\(x\approx52.6\) mm, \(y\approx38.2\) mm, \(A = 54^{\circ}\)