Question

in the figure below, △xyz ~ △acb. find cos b, sin b, and tan b. round your answers to the nearest hundredth.

Explanation:

Step1: Use similarity of triangles

Since \(\triangle XYZ\sim\triangle ACB\), the ratios of corresponding - sides are equal. Also, in right - triangle trigonometry, for right - triangle \(ACB\) with right - angle at \(A\), \(\cos B=\frac{AB}{CB}\), \(\sin B = \frac{AC}{CB}\), and \(\tan B=\frac{AC}{AB}\). The corresponding sides of similar triangles give us the same trigonometric ratios. In \(\triangle XYZ\), if we consider the angle \(Y\) (corresponding to angle \(B\)), \(\cos Y=\frac{XY}{YZ}\), \(\sin Y=\frac{XZ}{YZ}\), \(\tan Y=\frac{XZ}{XY}\).

Step2: Calculate \(\cos B\)

\(\cos B=\frac{XY}{YZ}=\frac{18}{20.4}\approx0.88\)

Step3: Calculate \(\sin B\)

\(\sin B=\frac{XZ}{YZ}=\frac{9.6}{20.4}\approx0.47\)

Step4: Calculate \(\tan B\)

\(\tan B=\frac{XZ}{XY}=\frac{9.6}{18}\approx0.53\)

Answer:

\(\cos B\approx0.88\)
\(\sin B\approx0.47\)
\(\tan B\approx0.53\)