Question

question 17 (mandatory) (1 point)
determine the measure of the angle \\( \theta \\) to the nearest degree.

\\( \bigcirc \\) a) \\( \theta=64^{\circ} \\)
\\( \bigcirc \\) b) \\( \theta=68^{\circ} \\)
\\( \bigcirc \\) c) \\( \theta=66^{\circ} \\)
\\( \bigcirc \\) d) \\( \theta=62^{\circ} \\)

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 \(\frac{2.54}{\sin43^{\circ}}=\frac{3.46}{\sin\theta}\).

Step2: Solve for \(\sin\theta\)

Cross - multiply: \(2.54\sin\theta=3.46\sin43^{\circ}\). Since \(\sin43^{\circ}\approx0.682\), then \(2.54\sin\theta = 3.46\times0.682\). \(2.54\sin\theta=2.36\). So, \(\sin\theta=\frac{2.36}{2.54}\approx0.929\).

Step3: Find \(\theta\)

Take the inverse - sine: \(\theta=\sin^{- 1}(0.929)\). Using a calculator, \(\theta\approx68^{\circ}\).

Answer:

B. \(\theta = 68^{\circ}\)