MAT1793-ABED_1: Multiple Step Problems

Constructing Knowledge

Some problems require multiple steps and often involve the use of multiple right triangles. It is sometimes useful to state the measurement you are trying to find and then decide what additional information is required to determine that measurement. A good habit is to add any calculated values to the diagram as you go.

Multimedia

A video describing how to solve multiple step trigonometric problems is provided.

EXAMPLE 1

Determine the value of θ, to the nearest degree.

Solution

Step 1: Create a plan.

To solve for θ, either side a or d must be used with the hypotenuse. Currently both a and d are unknown.

Information is not readily available to solve for side length d. However, side length a can be determined using Pythagorean theorem.

Once a is known, θ can be calculated.

Step 2: Solve for side length a.

\(\begin{align} a^2+b^2&=c^2 \\ \\ a^2+5^2&=7^2 \\ \\ a^2+25&=49 \\ \\ a^2+25-25&=49-25 \\ \\ a^2&=24 \\ \\ \sqrt{a^2}&=\sqrt{24} \\ \\ a&=4.9 \\ \end{align}\)

Label the new measurement on the diagram.

Step 3: Solve for θ.

\(\begin{align} \text{sin}\,\theta&=\frac{\text{length opposite}\,\theta}{\text{hypotenuse}} \\ \\ \text{sin}\,\theta&=\frac{4.9}{13} \\ \\ \theta&=\text{sin}^{-1}\left(\frac{4.9}{13}\right) \\ \\ \theta&=22° \end{align}\)

The measure of angle θ is 22°.