Digital Lesson
Trigonometric
Applications and Models
Trigonometric Functions on a Calculator
Example 1: Calculate sin 40.
Set the calculator in degree mode.
Calculator keystrokes: sin 40 =
Display: 0.6427876
Example 2: Calculate sec 40.
Calculator keystrokes: 1  cos 40 =
Display: 1.3054072
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2
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.
a
θ
a
a
• an angle  and a side a,
b
a
• or two sides, a and b.
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θ
θ
a
b
a
b
3
Solving A Right Triangle Given an Angle and a Side
Solve the right triangle.
The third angle is 60, the complement
of 30. Use the values of the
trigonometric functions of 30o.
5
30○
1
opp
Since = sin 30 =
= 5 , it follows that hyp = 10.
hyp hyp
2
To get the last side, note that
3 2 = cos 30 = adj ;
10
therefore, adj = 5 3.
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10
60○
5
30○
5 3
4
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.
x
Post B
Post A
Use a calculator to find
tan 73o = 3.27.
100 ft.
○
x
opp
73
3.27 = tan 73=
=
adj
100
It follows that x = 327.
The distance across the river from post A to post B is 327 feet.
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5
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1 0.25 =
Display: 75.22487
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6
Solving a Right Triangle Given Two Sides
Solve the right triangle shown.
Solve for the hypotenuse:
hyp2 = 62 + 52 = 61
hyp = 61 = 7.8102496
Solve for  :
tan  = opp = 5 and  = tan-1( 5 ).
adj
6
6
5
θ
6
61 50.2○
5
39.8○
6
Calculator Keystrokes: (SHIFT) tan1 ( 5  6 )
Display: 39.805571
Subtract to calculate the third angle:
90 39.805571 = 50.194428.
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7
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.
line of sight
object
angle of elevation
horizontal
observer
When an observer is looking downward, the angle formed
by a horizontal line and the line of sight is called the:
horizontal
angle of depression
line of sight
object
observer
angle of depression.
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8
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
cliff
42 m
horizontal
16○ angle of depression
line of sight
16○
d
ship
42
= 146.47.
tan 16
The ship is 146 m from the base of the cliff.
d=
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9
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
14
16
sin  =
= 0.875
14
16
θ
Next use the inverse sine function to find .
 = sin1(0.875) = 61.044975
The angle formed by the ladder and the ground is about 61.
The painter’s plan is unsafe!
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