Question

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

a) θ = 39°
b) θ = 41°
c) θ = 47°
d) θ = 36°

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). First, find the length of side \(AC\) (let \(AB = c = 11.1\), \(BC=a = 10.0\), \(\angle C=41^{\circ}\)). Then find \(\angle A\) (denoted as \(\theta\)) using the Law of Sines \(\frac{\sin A}{a}=\frac{\sin C}{c}\).

Step2: Apply the Law of Sines

We know \(a = 10.0\), \(c = 11.1\), \(\angle C = 41^{\circ}\). From the Law of Sines \(\sin\theta=\frac{a\sin C}{c}\). Substitute the values: \(\sin\theta=\frac{10\times\sin41^{\circ}}{11.1}\). Calculate \(\sin41^{\circ}\approx0.6561\). Then \(\sin\theta=\frac{10\times0.6561}{11.1}\approx0.5911\).

Step3: Find the angle

Take the inverse - sine of \(0.5911\), \(\theta=\sin^{- 1}(0.5911)\approx36.2^{\circ}\approx36^{\circ}\)

Answer:

d) \(\theta = 36^{\circ}\)