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 = 10.5 ), ( angle a = 73^{circ} )
( a=)
( c=)
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Explanation:

Step1: Find angle \( B \)

In a right - triangle \( A + B+C=180^{\circ}\), and \( C = 90^{\circ}\). So \( B=180^{\circ}-(A + C)=180^{\circ}-(73^{\circ}+90^{\circ}) = 17^{\circ}\)

Step2: Find side \( a \)

We know that \(\tan A=\frac{a}{b}\). Given \( b = 10.5\) and \(A = 73^{\circ}\), then \(a=b\tan A\).
\(a = 10.5\times\tan(73^{\circ})\approx10.5\times3.2709 = 34.3445\)

Step3: Find side \( c \)

We know that \(\cos A=\frac{b}{c}\), then \(c=\frac{b}{\cos A}\).
Since \(b = 10.5\) and \(A = 73^{\circ}\), \(c=\frac{10.5}{\cos(73^{\circ})}\approx\frac{10.5}{0.2924}\approx35.9108\)

Answer:

\(a\approx34.3445\), \(c\approx35.9108\)