Question

solve for the requested variable based on the given information. round your answers to the nearest hundredth

1. a = 38°, b = 10, solve for c
2. b = 88°, b = 2, solve for a
3. a = 44°, c = 12, solve for b
4. b = 71°, a = 18, solve for c

Explanation:

Step1: Recall cosine formula for right - triangle

In right - triangle \(ABC\) with right - angle at \(C\), \(\cos A=\frac{b}{c}\). Given \(A = 38^{\circ}\) and \(b = 10\), we can solve for \(c\) as \(c=\frac{b}{\cos A}\).
\[c=\frac{10}{\cos38^{\circ}}\]
Since \(\cos38^{\circ}\approx0.7880\), then \(c=\frac{10}{0.7880}\approx12.69\)

Step2: Recall sine formula for right - triangle

In right - triangle \(ABC\) with right - angle at \(C\), \(\sin B=\frac{b}{c}\) and \(\tan B=\frac{b}{a}\). Given \(B = 88^{\circ}\) and \(b = 2\), from \(\tan B=\frac{b}{a}\), we have \(a=\frac{b}{\tan B}\)
\[a=\frac{2}{\tan88^{\circ}}\]
Since \(\tan88^{\circ}\approx28.6363\), then \(a=\frac{2}{28.6363}\approx0.07\)

Step3: Recall cosine formula for right - triangle

In right - triangle \(ABC\) with right - angle at \(C\), \(\cos A=\frac{b}{c}\). Given \(A = 44^{\circ}\) and \(c = 12\), we can solve for \(b\) as \(b = c\cos A\)
\[b = 12\times\cos44^{\circ}\]
Since \(\cos44^{\circ}\approx0.7193\), then \(b=12\times0.7193 = 8.63\)

Step4: Recall cosine formula for right - triangle

In right - triangle \(ABC\) with right - angle at \(C\), \(\cos B=\frac{a}{c}\), so \(c=\frac{a}{\cos B}\). Given \(B = 71^{\circ}\) and \(a = 18\), we have
\[c=\frac{18}{\cos71^{\circ}}\]
Since \(\cos71^{\circ}\approx0.3256\), then \(c=\frac{18}{0.3256}\approx55.28\)

Answer:

1. \(c\approx12.69\)
2. \(a\approx0.07\)
3. \(b\approx8.63\)
4. \(c\approx55.28\)