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3.3 Trigonometry
Name ________________________
The first part of this set of lecture notes will be considered an activity.
Trigonometry is the study of right triangles. More specifically, it is the study of how angles
and sides relate to each other. Before we make use of these relationships we are going to first
determine them. You will need a calculator, protractor, ruler, pencil and (of course) an eraser.
Fill out Table 1 by measuring each side to the nearest 10th of a centimeter. Measure
each angle to the nearest degree.
#1
NOTE: In these exercises we will use the protocol that angle C is the right angle and “c” is the
hypotenuse, side “a” is opposite angle A, and side “b” is opposite angle B.
Triangle 1
Triangle 2
A
A
c
b
C
B
a
c
b
C
B
a
1
The results are not going to be PERFECT because of errors that accumulate due to drawing
with computer tools and reproduction differences.
TABLE #1
Triangle 1
Triangle 2
Side a (cm)
Side b (cm)
Side c (cm)
Angle A
Angle B
Angle C
*Ratio of a to c
a c
*Ratio of b to c
b c
*Ratio of b to a
ba
* I am looking for a decimal answer rounded off to two decimal places.
1) If all went well, you should find the two triangles to be “similar.” That means, they are
the same shape, just a different size. Angles A, B and C should be the same (or very,
very, very close – within a degree or two) in both triangles. If this is not true, you should
go back and check.
2) Also, given that the triangles ARE similar, the ratios you found (not the values for a, b
and c) should be the same (or very, very, very close) as well.
If your values are “way off,” go back and measure again.
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#2
It is important for you to know that it is the measures of the given angles and sides that
drive the measures of the angles and sides that you
TABLE #2
don’t know. Let me explain by example.
GIVEN 2 ANGLES AND 1 SIDE
At the bottom of the page, draw a right triangle labeled
ABC with the following criteria:
Angle A
a) mC  90
b) mB  30
c) Let leg “a” = 6 cm (the side opposite
 A ).
Find the measure of angle C by actually measuring it. Fill
in Table 2.
Find the measure of the missing leg “b” – it is opposite
angle B. Fill in Table 2.
Find the measure of the hypotenuse “c” – it is opposite
angle C. Fill in Table 2.
Angle B
30o
Angle C
90o
Side (leg) a
6cm
Side (leg) b
Side (hyp) c
Check your answers by using the Pythagorean Theorem.
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#3
a)
At the bottom of the page, draw a right triangle labeled ABC with the following criteria:
mC  90
TABLE #3
SIDE ANGLE SIDE
b) Side a (opposite angle A) = 8 cm
Angle A
c) Side b (opposite angle B) = 9 cm
Angle B
Find the measure of angles B and C by actually measuring
them. Do they add to 90o? They should. Fill in Table 2.
Find the measure of side “c” (the hypotenuse) by actually
measuring it. Check your numbers with the Pythagorean
Theorem. Fill in Table 2.
Angle C
90o
Side (leg) a
8cm
Side (leg) b
9cm
Side (hyp) c
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The message is:
If you know
two angles and one side
or
two sides and one angle
then you can find the missing sides and missing angles.
This is where the trigonometry comes in.
 Recall in a previous table, we looked at some ratios.
 Remember, these ratios were driven by the angles in the triangle (and, of course, the
sides).
 Trigonometry uses a shortcut method of referring to these ratios. The shortcut
terminology is sine, cosine, and tangent. Each of these means a very specific ratio.
Look at the triangle below.
Find the angle Θ (which is the
same as angle B).
The side ADJACENT to Θ is “a.:
The side OPPOSITE Θ is “b.”
The HYPOTENUSE is “c.”
Consider angle A
(stand on it)
The ratio of
a/c is the
cosine of Θ
cos  
adjacent
a

hypotenuse c
cos A 
The ratio of
b/c is the
sine of Θ
sin  
opposite
b

hypotenuse c
sin A 
opposite b

adjacent a
tan A 
The ratio of
b/a is the
cosine of Θ
tan  
Now, do the same for angle A, use this column.
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Draw another right triangle. Label the sides r, s, t. Name the triangele RST. So we are all
looking at the same kind of triangle, make S the right angle. Fill out the table.
Angle R
Angle T
Find the side adjacent
_______________
_______________
Find the side opposite
_______________
_______________
Find the hypotenuse
_______________
_______________
Now, we’re ready to apply the trig functions to what they represent.
It all boils down to this:
Sine =
Opposite
Hypotenuse
Cosine =
SOH
Adjacent
Hypotenuse
–
Tangent =
CAH
–
Opposite
Adjacent
TOA
Pronounced: “so-cuh-toe-uh”
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I know it sounds kooky, but THIS you MUST memorize.
Practice:
Draw a right triangle – any right triangle.
Name the angles (you pick) and assign values to all 3 angles, remember, this is a right triangle.
Name the
a. adjacent side ___________
Someone in class needs to choose.
b. opposite side ___________
c. hypotenuse ___________
Fill in the measures of the angles. Then, use your calculator to fill in the rest of the table.
Round your answers off to the nearest tenth.
Angle ____ = _______°
cos _____ = ______
sin _____ = ______
tan _____ = ______
Angle ____ = _______°
cos _____ = ______
sin _____ = ______
tan _____ = ______
Angle ____ = _______°
cos _____ = ______
sin _____ = ______
tan _____ = ______
Need more explanation? Check out this website a student found. The explanations are VERY
straightforward. (Thanks, M.B.)
http://www.youtube.com/watch?v=T8B8Tcz8kog&feature=related
And this one from Khan Academy
http://www.khanacademy.org/math/trigonometry/basic-trigonometry
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http://www.khanacademy.org/math/trigonometry/basic-trigonometry
Sin =
Opp
Hyp
Cos=
Adj
Hyp
Tan =
Opp
Adj
Time to learn how to use your calculator for trig functions.
Example 1: Find the missing sides. Round to the hundredths.
We know two angles and one side (the hypotenuse.)
Find x (the side adjacent to 40 ).
Which function should we use?
Find y (the side opposite 40 ).
Which function should we use?
Do these numbers work using the Pythagorean Theorem?
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Sin =
Opp
Hyp
Cos=
Adj
Hyp
Tan =
Opp
Adj
Example 2: Find the missing sides to the nearest 16th inch.
We know two angles (90 and ?) and one side (leg).
Find w.
Which function should we use?
Find z.
Which function should we use?
Do these numbers work using the Pythagorean Theorem?
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Sin =
Opp
Hyp
Cos=
Adj
Hyp
Tan =
Opp
Adj
Example 3: Find the horizontal and vertical
distance to drill holes in locations A and B. We
are looking for coordinates (x,y) for each
location.
Start with A.
B?
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cos
Recall that we use these functions
angles (90 and ?) and one side.
Sin =
Opp
Hyp
Cos=
Adj
Hyp
sin
tan when you know two
Opp
Adj
Tan =
Now, we will look at three new functions. All are related to the three trig functions we know.
These new functions are called INVERSE TRIG functions.
They look like this:
cos 1
sin 1
tan 1
We use these functions when we are looking for an ANGLE and we KNOW two sides.
Sin 1
Opp

Hyp
Cos 1
Adj

Hyp
Tan 1
Opp

Adj
Check out these videos:
http://youtu.be/pZDsiOSUznk
Example 4: Find angles A and B in a roof sloped at 7/12. Round to the nearest tenth place.
A?
B?
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Sin 1
Opp

Hyp
Cos 1
Adj

Hyp
Tan 1
Opp

Adj
Example 5: Finding the angles in the steel tubing.
Welders often use angled braces to add strength and rigidity to
a weld. Find angles A and B and the length of the tubing, from
long point to long point, from the given dimensions.
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Homework: You are assigned all of the problems from 1 - 23. Take a look at problems 24
– 27. If you are in the manufacturing field, you REALLY need to be able to do these. I will
give you an additional 2 points each (to be added to your next test), if you do them
correctly.
Practice 1: Figure
out the angles and
lengths in a regular
pentagon
Q1. Find the length
of one side
Q2. Find the total
height
Q3. Find the total
width
Q4. Find the length
of a diagonal
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Practice 1 Solved: Figure out the angles and lengths in a
regular pentagon
Q1.
Q2.
Q3.
Q4.
Find the length of one side
Find the total height
Find the total width
Find the length of a diagonal (????)
Three triangles (totally 540⁰), divided by 5
angles = 108⁰ for each outside angle.
The interior angles as shown below are each 36⁰.
Q1: length of one side (2 times ”a”)
a
cos 54 
12 Length of one side is 14.10cm
12 cos 54  a
c
a  7.05
36⁰
12cm
b
54⁰
54⁰
a
Q2: total height “b” plus the radius (12cm).
9.71  12
b
sin 54 
12
21.71
12sin 54  b
Total height is 21.71cm.
b  9.71
Q3: total width (2 times “c”)
12cm
c
12
12sin 72  c
c  11.41
sin 72 
c
72⁰
Total width is 11.41cm.
Q4: length of a diagonal
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Practice 2: Figure out the
horizontal and vertical
distance to each point.
Q1. Point A
Q2. Point B
Q3. Point C
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Practice 2 Solved: Figure out the
horizontal and vertical distance to each
point. Round to the hundredth.
Q1. Point A
Q2. Point B
Q3. Point C
Point A. Focus on a right triangle that can be drawn
which includes point A.
If we can find the measure of angle 1,
we can find the lengths of the two
sides (a and b).
A
46mm
a
1
b
First find the angular distance between each of the
points. 360  7  51.43 Therefore, m1  90  51.43
m1  38.57
b
a
cos 38.57 
sin 38.57 
46
46
Thus, the horizontal and vertical distance to point A is
46
cos
38.57
b
46sin 38.57  a
(35.96mm, 28.68mm). Check with Pythagorean theorem.
b  35.96mm
a  28.68mm
Point B. Focus on a right triangle that can be drawn which includes point B.
Find the length of side c.
m2  angle between points - m1
c
sin12.86 
m2  51.43 - 38.57
46
m2  12.86
46sin12.86  c
Find the length of side d.
d
cos12.86 
46
46 cos12.86  d
d  44.85mm
A
46mm
1
c
angle 2
c  10.24mm
46mm
side d
Thus, the horizontal and vertical distance to point B is
(44.85mm, -10.24mm). Check with Pythagorean theorem.
Why is the vertical position negative?????
Point C. Focus on a right triangle that can be drawn with includes point C.
Find the length of side f.
Find the length of side g.
m3  3  (51.43) - 90
g
f
m3  64.29
sin 64.29 
cos 64.29 
46
46
46sin 64.29  g
46 cos 64.29  f
f  19.96mm
g  41.45mm
angle 3
f
g
46mm
Thus, the horizontal and vertical distance to point C is (-19.96mm, -41.45mm).
Check with Pythagorean theorem.
Why are both the horizontal and vertical positions negative?????
C
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Practice 3. Figure out the horizontal and vertical distance
to each point.
Q1. Point A Q4. Point D
Q2. Point B Q5. Point E
Q3. Point C Q6. Graph the points on graph paper
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d
Practice 3 Solved. Figure out the horizontal and
vertical distance to each point.
Q1. Point A Q4. Point D
Q2. Point B Q5. Point E
Q3. Point C Q6. Graph the points on graph paper
c
E
C
b
b
side b
108⁰
side a
(5.56”, 0)
b  17.12"
180-108=72⁰
Point B. Add 18” to the value from point A.
18  5.56  23.56
(23.56”, 0)
Point D. First find sides “c” and “d.”
side d
18⁰
18”
B
a
Point E. Use the same triangle as for point A.
Find the length of side b.
(0, 17.12”)
b
sin 72 
18
18sin 72  b
a
18
18cos 72  a
a  5.56"
cos 72 
18”
108⁰
a A
Find the length of side a.
18⁰
54⁰
d
c
Point A. Determine the measure of each angle in a pentagon.
Three triangles (totally 540⁰), divided by 5
angles = 108⁰ each.
side c
D
Use Pythagorean Theorem to check.
Point C. Add “a” to the value from point B.
go get the horizontal. Use “b” for the vertical.
5.56  23.56
29.12
(29.12”, 17.12”)
c
18
18cos 54  c
c  10.58
cos 54 
d
18
18sin 54  d
d  14.56
sin 54 
The horizontal for point D is “d” (14.56in).
The vertical for point D is “b” plus “c” (17.12+10.58=27.7in).
(14.56in, 27.7in)
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Section 3.3: Trigonometry
Section 3.3:
1. A = 85.2o
B = 4.8o
2. guy wire ≈ 30.6 ft and d ≈ 14.4 ft
3. 41.4o
4. 37.5o
5.
5”
11 16
6. 76
24.
Bend Allowance (B)
Set Back (D)
.962 mm
.676 mm
3”
38
o
7”
D = 132
3”
or 11’
16
-
3”
16
8. W = 15.7 m
H = 8.7 m
9. A = 132.1o, B = 47.9o, C = 42.1o, D = 18.9o,
E = 411.6 m, F = 268.7 m, G = 199.6 m
10. H = D = 33
3”
216
7”
7. C = 236 16 or 19’ - 816
11.
12.
13.
14.
23. 37o
15”
or
16
2’ - 9
25. a) 4"
b) 7.35"
26. a) 5.482"
b) 9.781"
27. a) 4.515 cm
b) 6.495 cm
15”
16
849 mm
x = 25.7 cm, y = 8.3 cm
x = 125.8 mm, y = -173.1 mm
A = 32.9o
B = 57.1o
1”
8
11”
,
16
h = 13
15. x = 2
y=1
7”
8
9”
16. 2 16
17. A = (96.5 mm, 17.0 mm)
B = (-63.0 mm, 75.1 mm)
C = (-33.5 mm, -92.1 mm)
D = (84.9 mm, -49.0 mm)
18. A = 25.5o
B = 64.5o
C = 232.6 cm
19. A = 33.7o
B = 56.3o
3”
4
5”
8
C = 81
D=6
20. L ≈ 5.56
T ≈ 3,336 ft-lbs (using the rounded value for L)
21. 45
1”
,
2
43, 40
1”
,
2
38, 35
1”
2
22. 1248” (1249” if you don’t round until the end)
19