1. Trigonometric Applications and Models
5.7
Trigonometric Applications and Models

2. Trigonometric Functions on a Calculator
Example 1: Calculate sin 40.
Set the calculator in degree mode. Calculator keystrokes: sin 40 =
Display:
Example 2: Calculate sec 40.
Calculator keystrokes: 1  cos 40 =
Display:
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Trigonometric Functions on a Calculator

3. Solving Right Triangles
Solving a right triangle means to find the lengths of the sides and the measures of the angles of a right triangle.
Some information is usually given. θ a
an angle  and a side a, b a
or two sides, a and b.
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Solving Right Triangles

4. Solving A Right Triangle Given an Angle and a Side
Solve the right triangle.
30○ 5
The third angle is 60, the complement of 30. Use the values of the trigonometric functions of 30o.
Since = sin 30 = = , it follows that hyp = 10.
hyp
opp 5
30○ 5
To get the last side, note that
= cos 30 = ;
therefore, adj = 10
60○
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Solving A Right Triangle Given an Angle and a Side

5. Example 1:
A bridge is to be constructed across a small river from post A to post B. A surveyor walks 100 feet due south of post A. She sights on both posts from this location and finds that the angle between the posts is 73. Find the distance across the river from post A to post B.
Post B
Post A
100 ft. x
73○
Use a calculator to find tan 73o = 3.27.
3.27 = tan 73= =
adj
opp
It follows that x = 327.
The distance across the river from post A to post B is 327 feet.
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Example 1: Application

6. Inverse Trigonometric Functions on a Calculator
Labels for sin1, cos1, and tan1 are usually written above the sin, cos, and tan keys.
Inverse functions are often accessed by using a key that maybe be labeled SHIFT, INV, or 2nd. Check the manual for your calculator.
Example: Find the acute angle  for which cos  = 0.25.
Calculator keystrokes: (SHIFT) cos =
Display:
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Inverse Trigonometric Functions on a Calculator

7. Solving a Right Triangle Given Two Sides
Solve the right triangle shown. θ 5
Solve for the hypotenuse:
hyp2 = = 61 6
hyp = = 6 5
50.2○
Solve for  :
tan  = = and  = tan-1( ).
adj
opp
39.8○
Calculator Keystrokes: (SHIFT) tan1 ( 5  6 ) Display: 
Subtract to calculate the third angle:   = .
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Solving a Right Triangle Given Two Sides

8. Angle of Elevation and Angle of Depression
When an observer is looking upward,
the angle formed by a horizontal line and the line of sight is called the:
angle of elevation.
observer
object
line of sight
horizontal
angle of elevation
When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the:
observer
object
line of sight
horizontal
angle of depression.
angle of depression
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Angle of Elevation and Angle of Depression

9. Example 2:
A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16. What is the distance from the ship to the base of the cliff?
observer
horizontal
16○
angle of depression
cliff 42 m
line of sight
16○
ship d
d = =
The ship is 146 m from the base of the cliff.
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Example 2: Application

10. Example 3:
A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60 angle of inclination. Does the painter’s plan satisfy the safety requirements for the use of the ladder?
ladder
house
sin  = = 0.875 16 14 θ
Next use the inverse sine function to find .
 = sin1(0.875) =
The angle formed by the ladder and the ground is about 61.
The painter’s plan is unsafe!
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Example 3: Application