Question

what is the measure of angle ( r ) in ( \triangle qrs ) shown below, given ( r = 10 ), ( q = 20 ), and ( q = 100^circ )? round the answer to the nearest tenth.
not drawn to scale
( 0.5^circ )
( 2.0^circ )
( 29.5^circ )
( 50.5^circ )

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin Q}{q}=\frac{\sin R}{r}\).
Substitute the given values: \(\frac{\sin 100^{\circ}}{20}=\frac{\sin R}{10}\).

Step2: Solve for \(\sin R\)

Cross - multiply: \(10\times\sin 100^{\circ}=20\times\sin R\).
Then \(\sin R=\frac{10\times\sin 100^{\circ}}{20}\).
Since \(\sin 100^{\circ}\approx0.9848\), we have \(\sin R=\frac{10\times0.9848}{20}= 0.4924\).

Step3: Find the measure of angle \(R\)

Take the inverse sine: \(R = \sin^{-1}(0.4924)\).
Using a calculator, \(R\approx29.5^{\circ}\).

Answer:

\(29.5^{\circ}\)