Exemplo 2 de Algebra and Trigonometry, 10.1 Non-right Triangles: Law of Sines

Solving an Oblique SSA Triangle

Problema e solução em inglês, como no livro original.

Solve the triangle in Figure 10 for the missing side and find the missing angle measures to the nearest tenth.

An oblique triangle with standard labels where side a is of length 6, side b is of length 8, and angle alpha is 35 degrees.

Figure 10

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
Use the Law of Sines to find angle $β$ and angle $γ,$ and then side $c .$ Solving for $β,$ we have the proportion

$\begin{array}{r} \frac{\sin α}{a} = \frac{\sin β}{b} \\ \frac{\sin (35°)}{6} = \frac{\sin β}{8} \\ \frac{8 \sin (35°)}{6} = \sin β \\ 0.7648 \approx \sin β \\ \sin^{- 1} (0.7648) \approx 49.9° \\ β \approx 49.9° \end{array}$
However, in the diagram, angle $β$ appears to be an obtuse angle and may be greater than 90°. How did we get an acute angle, and how do we find the measurement of $β ?$ Let’s investigate further. Dropping a perpendicular from $γ$ and viewing the triangle from a right angle perspective, we have Figure 11. It appears that there may be a second triangle that will fit the given criteria.

Figure 11

The angle supplementary to $β$ is approximately equal to 49.9°, which means that $β = 180° - 49.9° = 130.1° .$ (Remember that the sine function is positive in both the first and second quadrants.) Solving for $γ,$ we have
$γ = 180° - 35° - 130.1° \approx 14.9°$
We can then use these measurements to solve the other triangle. Since $γ^{'}$ is supplementary to the sum of $α^{'}$ and $β^{'},$ we have
$γ^{'} = 180° - 35° - 49.9° \approx 95.1°$
Now we need to find $c$ and $c^{'} .$

We have

$\begin{array}{l} \frac{c}{\sin (14.9°)} = \frac{6}{\sin (35°)} \\ c = \frac{6 \sin (14.9°)}{\sin (35°)} \approx 2.7 \end{array}$
Finally,

$\begin{array}{l} \frac{c^{'}}{\sin (95.1°)} = \frac{6}{\sin (35°)} \\ c^{'} = \frac{6 \sin (95.1°)}{\sin (35°)} \approx 10.4 \end{array}$
To summarize, there are two triangles with an angle of 35°, an adjacent side of 8, and an opposite side of 6, as shown in Figure 12.

Figure 12

However, we were looking for the values for the triangle with an obtuse angle $β .$ We can see them in the first triangle (a) in Figure 12.

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Given $α = 80°, a = 120,$ and $b = 121,$ find the missing side and angles. If there is more than one possible solution, show both.

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