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Exercises Trigonometry

Example 1 from Algebra and Trigonometry, 10.1 Non-right Triangles: Law of Sines

Solving for Two Unknown Sides and Angle of an AAS Triangle

Solve the triangle shown in Figure 7 to the nearest tenth.

An oblique triangle with standard labels. Angle alpha is 50 degrees, angle gamma is 30 degrees, and side a is of length 10. Side b is the horizontal base.

Figure 7

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
The three angles must add up to 180 degrees. From this, we can determine that
$\begin{array}{l} \begin{array}{l} β = 180° - 50° - 30° \end{array} \\ = 100° \end{array}$
To find an unknown side, we need to know the corresponding angle and a known ratio. We know that angle $α = 50°$ and its corresponding side $a = 10.$ We can use the following proportion from the Law of Sines to find the length of $c .$
$\begin{array}{l} \frac{\sin (50°)}{10} = \frac{\sin (30°)}{c} \\ c \frac{\sin (50°)}{10} = \sin (30°) & Multiply both sides by c . \\ c = \sin (30 °) \frac{10}{\sin (50°)} & Multiply by the reciprocal to isolate c . \\ c \approx 6.5 \end{array}$
Similarly, to solve for $b,$ we set up another proportion.

$\begin{array}{l} \begin{array}{l} \frac{\sin (50°)}{10} = \frac{\sin (100°)}{b} \end{array} \\ b \sin (50°) = 10 \sin (100°) & Multiply both sides by b . \\ b = \frac{10 \sin (100°)}{\sin (50°)} \begin{matrix} \end{matrix} & Multiply by the reciprocal to isolate b . \\ b \approx 12.9 \end{array}$
Therefore, the complete set of angles and sides is

$\begin{array}{l} \begin{array}{l} α = 50° a = 10 \end{array} \\ β = 100° b \approx 12.9 \\ γ = 30° c \approx 6.5 \end{array}$

Your turn

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

Solve the triangle shown in Figure 8 to the nearest tenth.

An oblique triangle with standard labels. Angle alpha is 98 degrees, angle gamma is 43 degrees, and side b is of length 22. Side b is the horizontal base.

Figure 8

How did it go?

PreviousUsing the Pythagorean Theorem to Model an Abstract Problem Next Solving an Oblique SSA Triangle

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.