Example 3 from Algebra and Trigonometry, 10.1 Non-right Triangles: Law of Sines

Solving for the Unknown Sides and Angles of a SSA Triangle

In the triangle shown in Figure 13, solve for the unknown side and angles. Round your answers to the nearest tenth.

An oblique triangle with standard labels. Side b is 9, side c is 12, and angle gamma is 85. Angle alpha, angle beta, and side a are unknown.

Figure 13

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
In choosing the pair of ratios from the Law of Sines to use, look at the information given. In this case, we know the angle $γ = 85°,$ and its corresponding side $c = 12,$ and we know side $b = 9.$ We will use this proportion to solve for $β .$
$\begin{array}{l} \frac{\sin (85°)}{12} = \frac{\sin β}{9} \begin{matrix} \end{matrix} & Isolate the unknown . \\ \frac{9 \sin (85°)}{12} = \sin β \end{array}$
To find $β,$ apply the inverse sine function. The inverse sine will produce a single result, but keep in mind that there may be two values for $β .$ It is important to verify the result, as there may be two viable solutions, only one solution (the usual case), or no solutions.
$\begin{array}{l} β = \sin^{- 1} (\frac{9 \sin (85°)}{12}) \\ β \approx \sin^{- 1} (0.7471) \\ β \approx 48.3° \end{array}$
In this case, if we subtract $β$ from 180°, we find that there may be a second possible solution. Thus, $β = 180° - 48.3° \approx 131.7° .$ To check the solution, subtract both angles, 131.7° and 85°, from 180°. This gives
$α = 180° - 85° - 131.7° \approx - 36.7°,$
which is impossible, and so $β \approx 48.3° .$
To find the remaining missing values, we calculate $α = 180° - 85° - 48.3° \approx 46.7° .$ Now, only side $a$ is needed. Use the Law of Sines to solve for $a$ by one of the proportions.
$\begin{array}{l} \begin{array}{l} \begin{array}{l} \frac{\sin (85 °)}{12} = \frac{\sin (46.7 °)}{a} \end{array} \end{array} \\ a \frac{\sin (85 °)}{12} = \sin (46.7 °) \\ a = \frac{12 \sin (46.7 °)}{\sin (85 °)} \approx 8.8 \end{array}$
The complete set of solutions for the given triangle is

$\begin{array}{l} \begin{array}{l} α \approx 46.7° a \approx 8.8 \end{array} \\ β \approx 48.3° b = 9 \\ γ = 85° c = 12 \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given $α = 80°, a = 100, b = 10,$ find the missing side and angles. If there is more than one possible solution, show both. Round your answers to the nearest tenth.

How did it go?

Solution

Your turn

Solution

Your turn