Question

3 find the measure of angle θ. round your answer to the nearest degree. (there is a triangle abc with angle at a being 82°, side ac = 4.1 cm, side bc = 6.0 cm, and angle at b is θ. also, some handwritten notes related to the law of sines: angle a is 82°, side a = 6.0, side b = 4.1, and equations like 6 / sin82° = 4.1 / sinb etc.)

Explanation:

Step1: Apply Law of Sines

The Law of Sines states that for a triangle with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, \(a = 6.0\) cm (opposite angle \(A = 82^\circ\)), \(b = 4.1\) cm (opposite angle \(B=\theta\)). So we have \(\frac{6.0}{\sin 82^\circ}=\frac{4.1}{\sin\theta}\).

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

Cross - multiply to get \(6.0\times\sin\theta=4.1\times\sin 82^\circ\). Then \(\sin\theta=\frac{4.1\times\sin 82^\circ}{6.0}\). Calculate \(\sin 82^\circ\approx0.9903\), so \(4.1\times0.9903\approx4.0602\). Then \(\sin\theta=\frac{4.0602}{6.0}\approx0.6767\).

Step3: Find \(\theta\)

Take the inverse sine of \(0.6767\): \(\theta=\sin^{- 1}(0.6767)\). Using a calculator, \(\theta\approx42.6^\circ\), and rounding to the nearest degree, \(\theta\approx43^\circ\).

Answer:

\(43^\circ\)