Question

18 multiple choice 2 points for the triangle shown below, find b if ∠b = 75° and c = 15. (round your answer to the nearest tenth.)

Explanation:

Step1: Identify the trigonometric ratio

Since we know an angle (\(\angle B = 75^{\circ}\)) and the hypotenuse (\(c = 15\)), and we need to find the side opposite to the right - angle (assuming it's a right - triangle, as it's a common trigonometry problem setup). Wait, no, if it's a right - triangle with right - angle at \(C\), then \(\sin B=\frac{b}{c}\) (SOH - CAH - TOA: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)).

Step2: Solve for \(b\)

We have the formula \(b = c\times\sin B\). Substitute \(c = 15\) and \(B = 75^{\circ}\).
\(\sin(75^{\circ})=\sin(45^{\circ}+ 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966\)
\(b = 15\times\sin(75^{\circ})\)
\(b=15\times0.966 = 14.49\approx14.5\)

Answer:

\(14.5\)