1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 5
Trigonometric Functions
5.8 Part 1 Applications of Trigonometric Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Solve a right triangle.
Solve problems involving bearings.
Model simple harmonic motion.

3. Solving Right Triangles
Solving a right triangle means finding the missing lengths of its sides and the measurements of its angles. We will label right triangles so that side a is opposite angle A, side b is opposite angle B, and side c, the hypotenuse, is opposite right angle C.
When solving a right triangle,
we will use the sine, cosine,
and tangent functions, rather
than their reciprocals.

4. Example: Solving a Right Triangle
Let A = 62.7° and a = Solve
the right triangle, rounding lengths
to two decimal places.
Find the other acute angle B.
Find one missing side. Side b using angle B and side a.

5. Example: Solving a Right Triangle - continued
Find the other missing side. Side c using angle A and side a.

6. Example – Solving a Right Triangle
Solve the right triangle shown for all unknown sides and angles.
Solution:
Because C = 90 , it follows that A + B = 90  and
B = 90  – 34.2  = 55.8.

7. To solve for a, use the fact that a = b tan A.
Example – Solution
To solve for a, use the fact that
a = b tan A.
So, a = 19.4 tan 34.2   Similarly, to solve for c, use the fact that 

8. Your Turn – Solving a Right Triangle
Find the remaining parts of the triangle.

9. Solving an Applied Trigonometry Problem
Draw a sketch, and label it with the given information. Label the quantity to be with a variable.
Use the sketch to write an equation relating the given quantities to the variable.
Solve the equation, and check that your answer makes sense.

10. Angles of Elevation and Depression
 = 
Angle of Depression
Angle of Elevation

11. Example: Finding a Side of a Right Triangle
From a point on level ground 80 feet from the base of the Eiffel Tower, the angle of elevation is 85.4°. Approximate the height of the Eiffel Tower to the nearest foot.
The height of the Eiffel Tower is
approximately 994 feet.

12. Example – Finding a Side of a Right Triangle
A safety regulation states that the maximum angle of elevation for a rescue ladder is 72. A fire department’s longest ladder is 110 feet. What is the maximum safe rescue height?
Solution:
A sketch is shown.
From the equation sin A = a / c, it follows that a = c sin A
Figure 4.79

13. Example 2 – Solution = 110 sin 72  104.6.
cont’d
= 110 sin 72 
So, the maximum safe rescue height is about feet above the height of the fire truck.

14. Your Turn:
The angle of elevation of the Sun is 35.1 ̊ at the instant it casts a shadow 789 feet long of a water tower. Use this information to calculate the height of the water tower.

15. Your Turn:
Bob is at the top of a vertical cliff, 80m high. He sees a boat out at sea. The angle of depression from Bob to the boat is 34o. How far from the cliff base is the boat?

16. Example: Finding an Angle of a Right Triangle
A guy wire is 13.8 yards long and is attached from the ground to a pole 6.7 yards above the ground. Find the angle, to the nearest tenth of a degree, that the wire makes with the ground.
The wire makes an angle of
approximately 29.0° with the ground.

17. Your Turn:
A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end, as shown. Find the angle of depression of the bottom of the pool.

18. Your Turn:
The length of the shadow of a tree m tall is m. Find the angle of elevation of the sun.

19. Example - Find of an Object in the Distance
Finding the height of a tree on a mountain.

20. Example - Find of an Object in the Distance
Finding the height of a tree on a mountain.

21. Example - Find of an Object in the Distance
Finding the height of a tree on a mountain.

22. Your Turn:
At a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 35 ̊, while the angle to the top is 53 ̊ . Find the height, s, of the smokestack alone.