Question

a right triangle has side ( a ) opposite angle ( a ), side ( b ) opposite angle ( b ), and side ( c ) as the hypotenuse (opposite right angle ( c )).
the length of one side and the measure of one angle are given. use a calculator to find the length of each remaining side to 4 decimal places.
( b = 20, angle b = 25^{circ} )
( a=)
( c=)
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Explanation:

Step1: Find angle \(A\)

In a right - triangle \(A + B+C=180^{\circ}\), since \(C = 90^{\circ}\), then \(A=180^{\circ}-(90^{\circ}+B)\).
Substitute \(B = 25^{\circ}\), so \(A=180^{\circ}-(90^{\circ}+25^{\circ})=65^{\circ}\).

Step2: Find side \(a\)

We know that \(\tan B=\frac{b}{a}\), so \(a=\frac{b}{\tan B}\).
Given \(b = 20\) and \(B = 25^{\circ}\), then \(a=\frac{20}{\tan(25^{\circ})}\).
Using a calculator, \(\tan(25^{\circ})\approx0.4663\), so \(a=\frac{20}{0.4663}\approx42.9080\).

Step3: Find side \(c\)

We know that \(\sin B=\frac{b}{c}\), so \(c=\frac{b}{\sin B}\).
Given \(b = 20\) and \(B = 25^{\circ}\), then \(\sin(25^{\circ})\approx0.4226\), so \(c=\frac{20}{0.4226}\approx47.3261\).

Answer:

\(a\approx42.9080\), \(c\approx47.3261\)