Question

policies
current attempt in progress
solve the triangle.
35°
5.0 in.
a
α
β
4.3 in.
( a = 2.6 ^ { circ }, \beta = 142.4 ^ { circ }, a = 0.4 mathrm { in. } ; a = 107.4 ^ { circ }, \beta = 37.6 ^ { circ }, a = 7.8 mathrm { in. } )
( a = 6.8 ^ { circ }, \beta = 138.2 ^ { circ }, a = 0.9 mathrm { in. } ; a = 103.2 ^ { circ }, \beta = 41.8 ^ { circ }, a = 7.3 mathrm { in. } )
( a = 10.8 ^ { circ }, \beta = 134.2 ^ { circ }, a = 1.3 mathrm { in. } ; a = 99.2 ^ { circ }, \beta = 45.8 ^ { circ }, a = 6.9 mathrm { in. } )
( a = 22.5 ^ { circ }, \beta = 122.5 ^ { circ }, a = 2.3 mathrm { in. } ; a = 87.5 ^ { circ }, \beta = 57.5 ^ { circ }, a = 5.9 mathrm { in. } )
no triangle with the given measurements.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin\alpha}{4.3}=\frac{\sin35^{\circ}}{5.0}\). So, \(\sin\alpha=\frac{4.3\times\sin35^{\circ}}{5.0}\).
Calculate \(\sin35^{\circ}\approx0.5736\), then \(\sin\alpha=\frac{4.3\times0.5736}{5.0}=\frac{2.4665}{5.0} = 0.4933\).
So, \(\alpha=\sin^{- 1}(0.4933)\approx29.5^{\circ}\) or \(\alpha = 180^{\circ}-29.5^{\circ}=150.5^{\circ}\). But if \(\alpha = 150.5^{\circ}\), then \(\alpha+35^{\circ}=185.5^{\circ}>180^{\circ}\), so we discard this value.

Another way:
We know that \(A = 35^{\circ}\), \(b = 5.0\), \(c = 4.3\)
By the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
Let's check each option:
For the first - option:
If \(\alpha = 2.6^{\circ}\), \(\beta=142.4^{\circ}\), then \(2.6^{\circ}+142.4^{\circ}+35^{\circ}=180^{\circ}\)
\(\frac{a}{\sin2.6^{\circ}}=\frac{5.0}{\sin142.4^{\circ}}\)
\(\sin142.4^{\circ}=\sin(180 - 37.6)^{\circ}=\sin37.6^{\circ}\approx0.6093\), \(\sin2.6^{\circ}\approx0.0454\)
\(a=\frac{5.0\times\sin2.6^{\circ}}{\sin142.4^{\circ}}=\frac{5\times0.0454}{0.6093}\approx0.4\)
If \(\alpha = 107.4^{\circ}\), \(\beta = 37.6^{\circ}\), \(107.4^{\circ}+37.6^{\circ}+35^{\circ}=180^{\circ}\)
\(\frac{a}{\sin107.4^{\circ}}=\frac{5.0}{\sin37.6^{\circ}}\)
\(\sin107.4^{\circ}=\sin(180 - 72.6)^{\circ}=\sin72.6^{\circ}\approx0.9544\)
\(a=\frac{5.0\times\sin107.4^{\circ}}{\sin37.6^{\circ}}=\frac{5\times0.9544}{0.6093}\approx7.8\)

Answer:

\(a = 2.6^{\circ},\beta = 142.4^{\circ},a = 0.4\) in.; \(a = 107.4^{\circ},\beta = 37.6^{\circ},a = 7.8\) in.