Question

question 7 (mandatory) (1 point) determine the measure of the angle θ to the nearest degree. a) θ=39° b) θ=47° c) θ=36° d) θ=41°

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In \(\triangle ABC\), we know \(BC = 10.0\) m, \(AB=11.1\) m, and \(\angle C = 41^{\circ}\). Let \(\angle A=\theta\), \(\angle B = 180^{\circ}-\theta - 41^{\circ}=139^{\circ}-\theta\), \(a = BC = 10.0\), \(c = AB=11.1\). By the Law of Sines \(\frac{BC}{\sin\theta}=\frac{AB}{\sin C}\).

Step2: Substitute values

Substitute \(BC = 10.0\), \(AB = 11.1\), and \(\angle C=41^{\circ}\) into \(\frac{BC}{\sin\theta}=\frac{AB}{\sin C}\). We get \(\sin\theta=\frac{BC\times\sin C}{AB}\). Plugging in the numbers: \(\sin\theta=\frac{10.0\times\sin(41^{\circ})}{11.1}\). Since \(\sin(41^{\circ})\approx0.656\), then \(\sin\theta=\frac{10.0\times0.656}{11.1}=\frac{6.56}{11.1}\approx0.591\).

Step3: Find \(\theta\)

Take the inverse - sine of \(0.591\). \(\theta=\sin^{- 1}(0.591)\approx36^{\circ}\) (using a calculator, \(\sin^{-1}(0.591)\) gives a value. When we check the options, and considering the accuracy of trigonometric calculations for such problems).

Answer:

C. \(\theta = 36^{\circ}\)