Question

question 4 (mandatory) (1 point)
determine the measure of the angle \\( \theta \\) to the nearest degree.

a) \\( \theta = 64 ^ { \circ } \\)
b) \\( \theta = 68 ^ { \circ } \\)
c) \\( \theta = 66 ^ { \circ } \\)
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 $\frac{2.54}{\sin 43^{\circ}}=\frac{3.46}{\sin\theta}$.

Step2: Solve for $\sin\theta$

Cross - multiply: $2.54\sin\theta=3.46\sin43^{\circ}$.
We know that $\sin43^{\circ}\approx0.682$, so $2.54\sin\theta = 3.46\times0.682$.
$2.54\sin\theta=2.36$.
$\sin\theta=\frac{2.36}{2.54}\approx0.929$.

Step3: Find $\theta$

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

Answer:

B. $\theta = 68^{\circ}$