Lesson 3: Solving Problems using Trigonometry - Solving Right Triangles Example 2


EXAMPLE 2

Solve the following triangle. Express angle measures to the nearest degree and lengths to the nearest tenth.



Solution


Step 1: Solve for the third angle

The two acute angles in a right triangle add to 90° (All three angles sum to 180° and the right angle is 90°).

\(\begin{align} y+71°&=90° \\ \\ y+71°-71°&=90°-71° \\ \\ y&=19° \end{align}\)

Step 2: Use a trigonometric ratio to determine one of the unknown side lengths from one of the known angle measures and the hypotenuse.

Side length w will be solved first. The sine ratio will be used as the length opposite to 71° is being solved for and the hypotenuse and 71° angle are known.

\(\begin{align} \text{sin}\,\theta&=\frac{\text{opp}}{\text{hyp}} \\ \\ \text{sin}\,71°&=\frac{w}{15} \\ \\ 15\times \text{sin}\,71°&=\frac{w}{\cancel{15}}\times \cancel{15} \\ \\ 14.2&=W \end{align}\)


Step 3: Solve for the third side

Side length z will be solved for next. The cosine ratio will be used as the length adjacent to 71° is being solved and the hypotenuse and 71° angle are known.

\(\begin{align} \text{cos}\,\theta&=\frac{\text{adj}}{\text{hyp}} \\ \\ \text{cos}\,71°&=\frac{z}{15} \\ \\ 15\times \text{cos}\,71°&=\frac{z}{\cancel{15}}\times \cancel{15} \\ \\ 4.9&=z \\ \end{align}\)

The Pythagorean theorem could also have been used to solve for the third side. However, where possible, it is always a good idea to use information given rather that calculated values (in case an error was made).

Step 4: Label the diagram with all side lengths and angle measures.




Now, it is your turn! Complete the questions in your Chapter 4, Lesson 3 Practice Makes Perfect that refer to Solving Right Triangles.



2013 © Alberta Distance Learning Centre