Question

standard 14 homework
score: 8/30 answered: 6/18
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question 7
suppose ( a = 12 ) and ( a = 55 ^ { circ } ) in the right triangle shown above. find each of the following values. round your answers to one decimal place.
( b = )
( c = )
( b = ) degrees
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Explanation:

Step1: Find \(b\) using \(\tan\) function

In a right - triangle, \(\tan A=\frac{a}{b}\). Given \(a = 12\) and \(A=55^{\circ}\), we can solve for \(b\) as \(b=\frac{a}{\tan A}\).
Substitute \(a = 12\) and \(A = 55^{\circ}\) into the formula: \(b=\frac{12}{\tan(55^{\circ})}\).
Since \(\tan(55^{\circ})\approx1.4281\), then \(b=\frac{12}{1.4281}\approx8.4\).

Step2: Find \(c\) using \(\sin\) function

In a right - triangle, \(\sin A=\frac{a}{c}\). So, \(c=\frac{a}{\sin A}\).
Substitute \(a = 12\) and \(A = 55^{\circ}\) into the formula: \(c=\frac{12}{\sin(55^{\circ})}\).
Since \(\sin(55^{\circ})\approx0.8192\), then \(c=\frac{12}{0.8192}\approx14.7\).

Step3: Find \(B\) using the angle sum property of a triangle

In a triangle, the sum of angles is \(180^{\circ}\). For a right - triangle (\(90^{\circ}+A + B=180^{\circ}\)).
So, \(B=90^{\circ}-A\).
Substitute \(A = 55^{\circ}\) into the formula: \(B = 90^{\circ}-55^{\circ}=35^{\circ}\).

Answer:

\(b = 8.4\), \(c = 14.7\), \(B = 35.0\) degrees