Lesson 3: Solving Problems using Trigonometry - Multiple Step Problems Example 2

   Constructing Knowledge

Some problems will refer to an angle of depression or an angle of elevation.


Angle of Depression
The angle between the horizontal and an observer’s line of sight to an object below the horizontal

Angle of Elevation
The angle between the horizontal and an observer’s line of sight to an object above the horizontal

   Multimedia

A video of Example 2 is provided.


EXAMPLE 2

Two buildings are 10 m apart. From the top of the shorter building, the angle of elevation to the top of the taller building is 61°, and the angle of depression to the base of the taller building is 72°. How tall, to the nearest tenth of a metre, is each building?

Solution


Step 1: Begin by sketching a diagram to represent the situation.



Step 2: Create a plan.

Let b represent the height of the short, grey building and let a + b represent the height of the tall, brown building. Both a and b can be determined using the tangent ratio.

Step 3: Solve for a.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,61°&=\frac{a}{10\,\text{m}} \\ \\ 10\,\text{m}\times \text{tan}\,61°&=\frac{a}{\cancel{10\,\text{m}}}\times \cancel{10\,\text{m}} \\ \\ 18.0\,\text{m}&=a \\ \end{align}\)

Step 4: Solve for b.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,72°&=\frac{b}{10\,\text{m}} \\ \\ 10\,\text{m}\times \text{tan}\,72°&=\frac{b}{\cancel{10\,\text{m}}}\times \cancel{10\,\text{m}} \\ \\ 30.8\,\text{m}&=b \\ \end{align}\)

Step 5: Complete the calculation required to answer the question asked.

The short building is 30.8 metres tall.

The tall building's height needs to be calculated by adding a and b.

tall building height = a + b
= 18.0m + 30.8m
= 48.8m

The brown building is 48.8 metres tall.


Now, it is your turn! Complete the questions in your Chapter 4, Lesson 3 Practice Makes Perfect that refer to Multiple Step Problems.



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