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Chapter 7
Trigonometry
Math 1202
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Things to get you started
Trigonometry is a branch of Mathematics that allows you to determine the measures of sides
and angles in a triangle. In this course we will look at right triangles.
Labeling Triangles
 Angles are always indicated by capital letters
 Sides are indicated with lower case letters.
 The angles and their corresponding sides use the same letter.
A
Side AB same as Side c
Side BC same as Side a
Side AC same as Side b
c
b
a
C
B
Basics
 There are three sides to a triangle which can be labelled the hypotenuse, opposite and
adjacent sides.
 The hypotenuse will never change in a right triangle
 The opposite or adjacent side will depend on the angle you are referring to.
 The reference angle (usually , theta) will always be one of the acute angles.
 The opposite side is the side opposite the angle.
 The adjacent side will be the side that is left (It points to the right angle).
Ex) For the following triangle, state :
L
P
M
1)
Hypotenuse
2)
Side opposite M
3)
Side opposite L
4)
Side adjacent L
5)
Side adjacent M
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Assign Worksheet “Labeling Triangles”
Trigonometric Ratios:
sin  
cos 
tan  
opposite
hypotenuse
Read as “Sine of Theta”
adjacent
hypotenuse
Read as “Cosine of Theta”
opposite
adjacent
Read as “Tangent of Theta”
There are two ways to remember these ratios:
1)
SOHCAHTOA
2)
Some Old Hags, Can’t Afford Husbands, Till Old Age
Ex) Find the trigonometric ratios for angles A and B in the following triangle.
sin A
cosA
tan A
opp
hyp
5

13

adj
hyp
12

13


opp
adj
5

12
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sin B
cosB
tan B
opp
hyp
12

13

adj
hyp
5

13


opp
adj
12

5
Using Calculators to find Trigonometric Ratios
You can use any scientific calculator to calculate trigonometric ratios. Be sure your calculator is
in degree mode (not radian). Before calculators were readily available, trigonometric tables were
used. Always give values of trig ratios to four decimal places.
Ex 1) Use a calculator to find the following:
A)
sin 68º
B)
tan 47º
C)
cos 23º
Finding sides using Trigonometry
You can use trig ratios to find missing sides of a right triangle.
Steps
1)
Label the sides in terms of “Hyp”, “Opp” and “Adj” relative to the reference angle.
2)
For the given angle, determine which trig ratio can be used with the known information.
equation based on the trig ratio chosen.
4)
Solve the equation using cross multiplication to find the missing side.


Note
If the unknown value is on top, you multiply to find the missing value.
If the unknown value is on bottom, you divide to find the missing value.
Ex 2) Determine the value of x in each triangle:
A)
B)
C)
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Finding angles using Trigonometry
You can also find missing angles given the trig ratio. You must use sin
calculate missing angles. (You may have to press shift, inv…etc first)
1
, cos1 , tan 1 to
Ex 1) Use a calculator to find the value of θ:
A)
sin Ɵ = 0.3265
B)
cos Ɵ = 0.5468
C)
tan Ɵ = 1.0271
You can use trig ratios to find missing angles of a right triangle.
Steps
1)
Label the sides in terms of “Hyp”, “Opp” and “Adj” relative to the reference angle.
2)
For the given angle, determine which trig ratio can be used with the known information.
3)
Write an equation based on the trig ratio chosen.
4)
Solve the equation using sin
nearest degree)
1
, cos1 or tan 1 to find the missing angle. (Usually to
Ex 2) Determine the value of θ in each triangle:
A)
B)
C)
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4.5 Applications of Trigonometry
We have learned how to solve a right triangle given either;
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thelengths of any two sides
the length of one side and the measure of an acute angle
You can apply these skills to solve word problems involving right triangles. It helps to draw a
diagram of the situation so you can determine which trig ratio to use.
There are two types of angles
mentioned in word problems
involving trigonometry.
Angle of
Depression
Angle of
Elevation
Ex 1) An 8 m ladder is placed against the wall so that it makes an angle of elevation of 32̊ with
the ground. How far is the base of the ladder from the wall?
x
8
x  8 cos32o
x  6 .8
cos32o 
8m
32°
x
The base of the ladder is 6.8 m from the wall.
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Ex 2) A plane is flying at an altitude of 450 m. It lands at an angle of depression of 28̊. How far
does the plane fly before it lands?
28°
450 m
450
x
0 .4695 450

1
x
0 .4695x  450
450
x
0 .4695
x  958
sin 28 
28°
The plane flies 958 m.
Using Trigonometry to find missing sides in a right triangle