Question

question 10 (mandatory) (1 point)
determine the measure of the angle θ to the nearest degree.

a) θ = 68°
b) θ = 66°
c) θ = 62°
d) θ = 64°

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In \(\triangle ABC\), let \(a = 2.54\), \(b = 3.46\), \(C = 43^{\circ}\). Then \(\frac{2.54}{\sin B}=\frac{3.46}{\sin43^{\circ}}\). First, find \(\sin B\): \(\sin B=\frac{2.54\times\sin43^{\circ}}{3.46}\).

Step2: Calculate \(\sin B\)

We know that \(\sin43^{\circ}\approx0.682\). So \(\sin B=\frac{2.54\times0.682}{3.46}\). \(2.54\times0.682 = 1.73228\), and \(\frac{1.73228}{3.46}\approx0.5007\). Then \(B=\sin^{- 1}(0.5007)\approx30^{\circ}\).

Step3: Find \(\theta\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(\theta=180^{\circ}-43^{\circ}-B\). Substitute \(B\approx30^{\circ}\), \(\theta = 180^{\circ}-43^{\circ}-30^{\circ}=107^{\circ}\) (This is wrong. Let's start over. We should use \(\frac{\sin\theta}{3.46}=\frac{\sin43^{\circ}}{2.54}\). \(\sin\theta=\frac{3.46\times\sin43^{\circ}}{2.54}\). \(\sin43^{\circ}\approx0.682\), \(3.46\times0.682 = 2.36\), \(\frac{2.36}{2.54}\approx0.929\). \(\theta=\sin^{-1}(0.929)\approx68^{\circ}\)

Answer:

a) \(\theta = 68^{\circ}\)