Question

question 12 (mandatory) (1 point)
determine the measure of the angle θ to the nearest degree.

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question 13 (，mandatory)，(1 point)
in △abc, a = 5.4 m, b = 7.2 m, and c = 10.0 m. determine ∠c to the nearest degree.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). In \(\triangle ABC\) for Question 12, let \(a = 10.0\), \(b = 11.1\), \(c\) be the side opposite \(\theta\). Wait, no, better use the Law of Sines. The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, in \(\triangle ABC\) (Question 12), \(a = 10.0\), \(b = 11.1\), \(C = 41^{\circ}\). First, find \(\sin\theta\). By the Law of Sines: \(\frac{10.0}{\sin\theta}=\frac{11.1}{\sin41^{\circ}}\). Then \(\sin\theta=\frac{10.0\times\sin41^{\circ}}{11.1}\).

Step2: Calculate \(\sin41^{\circ}\)

We know that \(\sin41^{\circ}\approx0.656\). So \(\sin\theta=\frac{10.0\times0.656}{11.1}=\frac{6.56}{11.1}\approx0.591\).

Step3: Find \(\theta\)

\(\theta=\sin^{- 1}(0.591)\). Using a calculator, \(\theta\approx36.2^{\circ}\approx36^{\circ}\).

Answer:

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