Question

use the right triangle and the given information to solve the triangle.

a = 9; \beta = 27^{\circ}; find b, c, and \alpha.

b = \square (round to the nearest hundredth as needed.)

Explanation:

Step1: Recall trigonometric ratios

In a right triangle, \(\tan(\beta)=\frac{b}{a}\), where \(a\) is the adjacent side to \(\beta\), \(b\) is the opposite side to \(\beta\). We know \(a = 9\) and \(\beta=27^\circ\).

Step2: Solve for \(b\)

From \(\tan(\beta)=\frac{b}{a}\), we can rearrange to get \(b = a\times\tan(\beta)\). Substituting \(a = 9\) and \(\beta = 27^\circ\), we have \(b=9\times\tan(27^\circ)\). Calculating \(\tan(27^\circ)\approx0.5095\), then \(b\approx9\times0.5095 = 4.5855\approx4.59\).

Answer:

\(4.59\)