Question

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please view the following video before answering this question. lecture video, section 4.4, example 03
note: the figure is not drawn to scale.
in a triangle, ( a = 21 ), ( b = 13 ) and ( alpha = 126 ^ { circ } ). find all missing parts.
round the answers to the nearest integer.
( \beta = )
( gamma = )
( c = )

Explanation:

Step1: Use the Law of Sines to find \(\beta\)

The Law of Sines states that \(\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}\).
We know \(a = 21\), \(b = 13\), and \(\alpha=126^{\circ}\).
So, \(\sin\beta=\frac{b\sin\alpha}{a}\)
Substitute the values: \(\sin\beta=\frac{13\sin126^{\circ}}{21}\)
\(\sin126^{\circ}=\sin(180 - 54)^{\circ}=\sin54^{\circ}\approx0.809\)
\(\sin\beta=\frac{13\times0.809}{21}\approx\frac{10.517}{21}\approx0.501\)
\(\beta=\sin^{- 1}(0.501)\approx30^{\circ}\)

Step2: Use the angle - sum property of a triangle to find \(\gamma\)

The sum of angles in a triangle is \(180^{\circ}\), so \(\gamma=180^{\circ}-\alpha-\beta\)
Substitute \(\alpha = 126^{\circ}\) and \(\beta = 30^{\circ}\)
\(\gamma=180-(126 + 30)=24^{\circ}\)

Step3: Use the Law of Sines to find \(c\)

Again, by the Law of Sines \(\frac{a}{\sin\alpha}=\frac{c}{\sin\gamma}\)
\(c=\frac{a\sin\gamma}{\sin\alpha}\)
Substitute \(a = 21\), \(\sin\alpha\approx0.809\), \(\sin\gamma=\sin24^{\circ}\approx0.407\)
\(c=\frac{21\times0.407}{0.809}=\frac{8.547}{0.809}\approx10.56\approx11\)

Answer:

\(\beta = 30^{\circ}\), \(\gamma = 24^{\circ}\), \(c = 11\)