Question

consider the following triangle. apply the law of sines to the triangle to find the missing angle (in degrees). (round your answer to one decimal place.) for triangle pqr we have sin(θ) / --select-- = --select-- / --select--, so θ ≈ °.

Explanation:

Step1: Write the Law of Sines formula

The Law of Sines states that for a triangle with sides \(a\), \(b\), \(c\) and opposite - angles \(A\), \(B\), \(C\) respectively, \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). In triangle \(PQR\), let \(PQ = 8\), \(PR = 7\), \(\angle Q = 40^{\circ}\), and \(\angle R=\theta\). Then \(\frac{\sin\theta}{8}=\frac{\sin40^{\circ}}{7}\).

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

Cross - multiply the Law of Sines equation \(\frac{\sin\theta}{8}=\frac{\sin40^{\circ}}{7}\) to get \(\sin\theta=\frac{8\sin40^{\circ}}{7}\). We know that \(\sin40^{\circ}\approx0.6428\), so \(\sin\theta=\frac{8\times0.6428}{7}=\frac{5.1424}{7}\approx0.7346\).

Step3: Find the value of \(\theta\)

Since \(\sin\theta\approx0.7346\), then \(\theta=\sin^{- 1}(0.7346)\). Using a calculator, \(\theta\approx47.2^{\circ}\).

Answer:

\(\sin(\theta)=\frac{8\sin40^{\circ}}{7}\), \(\theta\approx47.2\)