# Introduction to Trigonometry - PowerPoint PPT Presentation

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## Introduction to Trigonometry

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### introduction to trigonometric ratios of tangent, sine and cosine. – PowerPoint PPT presentation

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Title: Introduction to Trigonometry

1
SOHCAHTOA
• passport to Trigonometry Land ...

CLICK TO CONTINUE
2
Oops! Not a triangle!
• ?
• I am sorry.
• I am not a triangle!
• Click to Review Try Again!

3
I am a Triangle!
• ?
• Good job!
• You know me!
• Click Ill take you BACK

4
What is a triangle?
• A triangle is a polygon made up of three
connected line segments in such a way that each
side is connected to the other two.

CLICK TO CONTINUE
5
Examples of triangles...
• All these polygons are tri-gons and commonly
called triangles

CLICK TO CONTINUE
6
These are not triangles...
• None of these is a triangle...

Can you tell why not?
CLICK TO CONTINUE
7
Try it yourself...
• Click on each one that IS a triangle?

CLICK TO CONTINUE
8
Right triangles
• A right triangle is a special triangle that has
one of its angles a right angle.
• You can tell it is a right triangle when when one
angle measures 900 or the right angle is marked
by a little square on the angle whose measure is
900.

CLICK TO CONTINUE
9
These are right triangles...
All right triangles
CLICK TO CONTINUE
10
These are not right triangles...
• These triangles are NOT right triangles. Explain
why not?

CLICK TO CONTINUE
11
Try it yourself ...
• Click on the triangle that is NOT a right
triangle?

CLICK TO CONTINUE
12
I am a right triangle
• ?
• I am sorry you did not recognize me as one.
• I am a right triangle!

Click to Review Try Again!
13
Correct That is not a right Triangle
• ?
• Well done!
• You know your right triangles well!
• Click Ill take you BACK

14
Hypotenuse
• The longest side of a right triangle is the
hypotenuse.
• The hypotenuse lies directly opposite the right
angle.
• The legs may be equal in length or one may be
longer than the other.

CLICK TO CONTINUE
15
Parts of a right triangle ...
• A right triangle has two legs and a hypotenuse...

CLICK TO CONTINUE
16
Try it yourself hypotenuse
• Click on the side that is the hypotenuse of the
right triangle.

CLICK TO CONTINUE
17
Try it yourself shorter leg
• Click on the side that is the shorter leg of the
right triangle.

CLICK TO CONTINUE
18
Try it yourself longer leg
• Click on the side that is the longer leg of the
right triangle.

CLICK TO CONTINUE
19
Correct Way to go!
• ?
• Excellent!
• Wise choice.
• Click Go BACK and CONTINUE

20
Incorrect Choice!
• ?
• I am sorry.
• You chose the wrong side!

21
Pythagorean Theorem...
• The right triangle has a special property, called
the Pythagorean Theorem, that can help us find
one side if we know the other two sides.

If the lengths of hypotenuse and legs are c, a
and b respectively, then c2 a2 b2
CLICK TO CONTINUE
22
Finding a side - hypotenuse
• Use the Pythagorean Theorem to find the length of
the missing side.

c2 a2 b2 x2 102 142 100 196
296 x sqrt(296) 17.2
CLICK TO CONTINUE
23
Finding a side leg
• Use the Pythagorean Theorem to find the length of
the missing side.

c2 a2 b2 152 102 x2 225 100
x2 x2 125 x sqrt(125) 11.18
CLICK TO CONTINUE
24
Try it yourself- hypotenuse
• Find the hypotenuse of the given right triangle
with the lengths of the legs known

A. 36
B. 10
C. 100
D. 64
CLICK TO CONTINUE
25
Try it yourself leg
• Find the leg of the given right triangle with the
lengths of the leg and hypotenuse known

A. 24
B. 6
C. 144
D. 12
CLICK TO CONTINUE
26
• ?
• Great job!
• You take after Pythagoras!
• Click Go BACK and CONTINUE

27
Oops not quite!
• ?
• I am sorry.
• Click to Review Try Again!

28
• In a right triangle, a given leg is called the
adjacent side or the opposite side, depending on
the reference acute angle.

Adjacent or opposite from an acute reference
angle refers only to legs and not the hypotenuse
CLICK TO CONTINUE
29
Opposite side (to an acute angle)
• In a right triangle, a given leg is called the
adjacent side or the opposite side, depending on
the reference acute angle.

leg2 is opposite to acute angle A leg1 is NOT
opposite to acute angle A
CLICK TO CONTINUE
30
Adjacent side (to an acute angle)
• In a right triangle, a given leg is called the
adjacent side or the opposite side, depending on
the reference acute angle.

leg1 is adjacent to acute angle A leg2 is NOT
CLICK TO CONTINUE
31
Try it yourself opposite
• Click on the side that is opposite to angle B.

CLICK TO CONTINUE
32
• Click on the side that is adjacent to angle B.

B
CLICK TO CONTINUE
33
Correct reference!
• ?
• Great job!
• You understood these references!
• Click Go BACK and CONTINUE

34
Not exactly
• ?
• I am sorry.
• Opposite is across from the angle.
• Adjacent is next to the angle.
• Hypotenuse is neither adjacent nor opposite.
• Click to Review Try Again!

35
Trigonometric ratios of an acute angles of a
right triangle
• The ratios of the sides of a right triangle have
special names.
• There are three basic ones we will consider
• sine
• cosine
• tangent

CLICK TO CONTINUE
36
The sine of an acute angle...
• Let the lengths of legs be a and b, and the
length of the hypotenuse be c. A is an acute
angle.

With reference to angle A, the ratio of the
length of the side opposite angle A to length of
the hypotenuse is defined as sine A
a/c Sine A is abbreviated Sin A. Thus, sin A
a/c.
CLICK TO CONTINUE
37
The cosine of an acute angle...
• Let the lengths of legs be a and b, and the
length of the hypotenuse be c. A is an acute
angle.

With reference to angle A, the ratio of the
length of the side adjacent to angle A to length
of the hypotenuse is defined as cosine A
b/c Cosine A is abbreviated to Cos
A. Thus, cos A b/c.
CLICK TO CONTINUE
38
The tangent of an acute angle...
• Let the lengths of legs be a and b, and the
length of the hypotenuse be c. A is an acute
angle.

With reference to angle A, the ratio of the
length of the side opposite to angle A to length
of the side adjacent to angle A is defined
as tangent A a/b tangent A is
abbreviated Tan A. Thus, tan A a/b
CLICK TO CONTINUE
39
SOHCAHTOA
S Sine O Opposite H Hypotenuse - C
Cosine A Adjacent H Hypotenuse - T
• This is a clever technique most people use to
remember these three basic trig ratios.
• SOH-CAH-TOA sounds strange? What if I told you it
was the ancient oriental queen who loved
Geometry? (not true!)

CLICK TO CONTINUE
40
Example sine, cosine and tangent ratios...
• Find the sine of the given angle. SOHCAHTOA

Sin B Opposite/Hypotenuse sin 53.10 16/20
4/5 0.80 Cos B Adjacent/Hypotenuse Cos 53.10
12/20 3/5 0.60 Tan B
Opposite/Adjacent Tan 53.10 16/12 5/3 1.67
CLICK TO CONTINUE
41
Try it yourself trig ratios...
• Find the value of sine, cosine, and tangent of
the given acute angle. SOHCAHTOA

CLICK TO CONTINUE
42
Thats the way!
• ?
• Outstanding !
• SOHCAHTOA would be proud of you.
• You may want to teach others!
• Click Go BACK and CONTINUE

43
You can do it Try again!
• ?
• I am sorry.
• Remember it is SOHCAHTOA all the way!
• You must have used the wrong ratio!
• Click to Review Try Again!

44
Angle or Side Lengths?
• Does the trig ratio depend on the size of the
angle or size of the side length?
• Let us consider similar triangles in our
investigation.

CLICK TO CONTINUE
45
Find the trig ratios of each...
Remember SOHCAHTOA!
• Compute the ratios and make a conjecture

sin 36.870 ? 0.6 6/10 0.6 9/15 0.6
cos 36.870 ? 0.8 8/10 0.8 12/15 0.8
tan 36.870 ¾ 0.75 6/8 0.75 9/12 0.75
Conjecture Trigonometric ratios are a property
of similarity (angles) and not of the length of
the sides of a right triangle.
CLICK TO CONTINUE
46
Computed Trig Ratios ...
• The trig ratios are used so often that technology
makes these values readily available in the form
of tables and on scientific calculators.
• We will now show you how to use your calculator
to find some trig ratios.
• Grab a scientific calculator and try it out.

CLICK TO CONTINUE
47
How to find trig values on calculator
• Each calculator brand may work a little
differently, but the results will be the same.
• Look for the trig functions on your calculator
sin, cos and tan
• select the trig ratio of your choice followed by
the angle in degrees and execute (enter).
• example sin 30 will display 0.5
• on some calculators you may have to type in the
angle first then the ratio
• example 30 sin will display 0.5

CLICK TO CONTINUE
48
Given an angle, use calculator to evaluate trig
ratio
• Use your calculator to verify that the sine,
cosine and tangent of the following angles are
correct (to 4 decimals)

Angle A sin A cos A tan A
45o Sin 45o 0.7071 Cos 45o 0.7071 Tan 45o 1.0000
60o Sin 60o 0.8660 Cos 60o 0.5000 Tan 60o 1.7321
30o Sin 30o 0.5000 Cos 30o 0.8660 Tan 30o 0.5774
82.5o Sin 82.5o 0.9914 Cos 82.5o 0.1305 Tan 82.5o 7.5958
CLICK TO CONTINUE
49
Try it yourself...
• Find the values of the following trig ratios to
four decimal places

CLICK TO CONTINUE
50
Going backwards finding angles when we know a
trig ratio
• We can use the reverse operation of a trig ratio
to find the angle with the known trig ratio (n/m)
• The inverse trig ratios are as follows
• Inverse of sin (n/m) is sin-1(n/m)
• Inverse of cos (n/m) is cos-1(n/m)
• Inverse of tan (n/m) is tan-1(n/m)

CLICK TO CONTINUE
51
Example inverse operation (sin A)
• Suppose we know the trig ratio and we want to
find the associated angle A.
• From SOHCAHTOA, we know that from the angle A, we
have the opposite side and the hypotenuse.
• Therefore the SOH part helps us to know that we
use sin A O/H 4/5
• The inverse is thus sin-1(4/5) A
• A Sin-1 (4/5) 53.13o

CLICK TO CONTINUE
52
Example inverse operation (cos B)
• Suppose we know the trig ratio and we want to
find the associated angle B.
• From SOHCAHTOA, we know that from the angle B, we
have the adjacent side and the hypotenuse.
• Therefore the CAH part helps us to know that we
use cos A A/H 4/5
• The inverse is thus cos-1(4/5) B
• B cos-1 (4/5) 36.87o

CLICK TO CONTINUE
53
Example inverse operation (tan A)
• Suppose we know the trig ratio and we want to
find the associated angle A.
• From SOHCAHTOA, we know that from the angle A, we
have the opposite side and the adjacent side.
• Therefore the TOA part helps us to know that we
use tan A O/A 4/3
• The inverse is thus tan-1(4/3) A
• A tan-1 (4/3) 53.13o

3
CLICK TO CONTINUE
54
Try it yourself(use inverse)
• Use a calculator to find the measure of the
angles A and B.

Use SOHCAHTOA as a guide to what ratio to use.
m?A ? A. 38.7 B. 51.3 C. 53.1
m?B ? A. 38.7 B. 51.3 C. 53.1
CLICK TO CONTINUE
55
Perfect!
• ?
• You make me smile!
• sin-1(x) is also referred to as arcsin(x)
• cos-1(x) is also referred to as arccos(x)
• tan-1(x) is also referred to as arctan(x)

Click Go BACK and CONTINUE
56
Looks like youre in trouble!
• ?
• I am sorry.
• Not to worry, I can help.
• Click to GO BACK, REVIEW TRY Again!

57
Finding the legs of a right
• Use trig ratios to find sides of a triangle.
Remember SOHCAHTOA!
• With reference to angle A,
• b is the length of side adjacent and
• a is the length of the side opposite the angle.
• the hypotenuse is given
• Strategy make an equation that uses only one leg
and the hypotenuse at a time.

CLICK TO CONTINUE
58
Finding the legs of a right
out the legs a and b in this case. (SOHCAHTOA!)

Using tangent tan A O/A Substituting values
from the tgriangle tan 30o a/b From the
calculator tan 30o 0.5774 Thus tan 30o a/b
0.5774 a/b And, a 0.5774(b) GETS YOU
STUCK!
CLICK TO CONTINUE
59
Finding the legs of a right
• Using sine ratio to find the leg of a triangle.
Remember SOHCAHTOA!

Using sine sin A O/H Substituting values
from the tgriangle Sin 30o a/12 From the
calculator sin 30o 0.5 Thus sin 30o a/12
0.5 a/12 And a 0.5(12) 6
CLICK TO CONTINUE
60
Finding the legs of a right
• Using the cosine ratio to find legs of a
triangle. Remember SOHCAHTOA!

Using cosine cos A A/H Substituting values
from the tgriangle cos 30o b/12 From the
calculator cos 30o 0.8660 Thus cos 30o b/12
0.866 b/12 And b 0.866(12) 10.39
CLICK TO CONTINUE
61
Try it yourself legs
• Find the lengths of the legs of the triangle and
the third angle. Choose the correct answer.

CLICK TO CONTINUE
62
Having trouble?
• ?
• I am sorry you are having problems.
• Not to worry, I can help.
• Click to GO BACK, REVIEW TRY Again!

63
You got it!
• ?
• Fantastic !
• You are on the right track.
• Click Go BACK and CONTINUE

64
Finding the hypotenuse ...
• Use trig ratios to find the hypotenuse of a
triangle. Remember SOHCAHTOA!
• With reference to angle A,
• b is the length of side adjacent and
• 12 is the length of the side opposite the angle.
• c is the hypotenuse
• Strategy make an equation that uses only one
unkown at a time.

CLICK TO CONTINUE
65
Finding the hypotenuse ...
• Use trig ratios to find the hypotenuse of a
triangle. Remember SOHCAHTOA!

Since 12 is opposite to the angle, we use the
sine ratio Sine A O/H Substituting values
from the tgriangle sin 30o 12/c From the
calculator sin 30o 0.5 Thus sin 30o 12/c
or 0.5 12/c c 12/0.5 24
CLICK TO CONTINUE
66
Try it yourself hypotenuse
• Find the lengths of the hypotenuse, leg b and the
third angle. Choose the best answer.

CLICK TO CONTINUE
67
Terrific!
• ?
• I am proud of your progress!
• Click Go BACK and CONTINUE

68
Help is a click away!
• ?
• Review and try again.
• PLEASE GO BACK, REVIEW TRY again.

69
Putting it all together
• We now have the tools we need to solve any right
triangle (to determine the lengths of each and
all sides and the angles, given minimal
information) Remember SOHCAHTOA!
• Typically you get two pieces of information
• One side length and one angle or
• Two sides lengths

CLICK TO CONTINUE
70
Solve the triangle (side angle)
• Given one side length and one angle, determine
the rest. Remember SOHCAHTOA !

Find measure of angle B and side lengths AC and
AB.
Since we know two angles (90 and 42) we can
determine the 3rd from the Triangle Angle Sum
Theorem m?B 1800 (900420) 480.
CLICK TO CONTINUE
71
Solve the triangle (side angle)
• Given one side length and one angle, determine
the rest. Remember SOHCAHTOA !

Strategy side with length 12 is opposite to
angle A. To find b, use tan A and to find c, use
sin A
sin A O/H sin 42 12/c 0.6691 12/c c
12/0.6691 c 17.93
tan A O/A tan 42 12/b 0.9004 12/b b
12/0.9004 b 13.33
CLICK TO CONTINUE
72
Try it yourself side angle
• Solve the triangle. Choose and check answer.

CLICK TO CONTINUE
73
Congratulations!
• ?
• Its fun when you get it!
• You are on the right track.
• Click Go BACK and CONTINUE

74
Sorry, not correct!
• ?
• I am sorry.
• Remember it is SOHCAHTOA all the way!
• You must have used the wrong ratio!
• Click to Review Try Again!

75
Solve the triangle (2 sides)
• Given two side lengths, solve the triangle.
Remember SOHCAHTOA !
• Strategy
• use Pythagorean Theorem to find the 3rd side
length, a.
• Use cosine ratio to find measure of angle A
• Use the Triangle Angle Sum Theorem to find the
measure of angle B.

CLICK TO CONTINUE
76
Solve the triangle (2 sides)
• Given two side lengths, solve the triangle.
Remember SOHCAHTOA !
• Using Pythagorean Theorem to find the 3rd side
length, a.
• c2 a2 b2 Pythagorean Theorem
• 172 a2 102 Substituting values
• 289 a2 100 Evaluating the squares
• a2 289-100 Addition property of
• a2 189 Simplifying
• a sqrt(189) 13.75 Taking square root.

CLICK TO CONTINUE
77
Solve the triangle (2 sides)
• Given two side lengths, solve the triangle.
Remember SOHCAHTOA !
• Using cosine ratio to find measure of angle A
• cos A A/H (the CAH part)
• cos A 10/17 (substituting values)
• m?A cos-1(10/17) (inverse of cosine)
• m?A 53.97o (Calculator)

CLICK TO CONTINUE
78
Try it yourself(2 sides)

CLICK TO CONTINUE
79
You have mastered lot!
• ?
• Congratulations !
• We are almost done!
• You did it again!
• Click Go BACK and CONTINUE

80
Sorry you got it wrong!
• ?
• I am sorry.
• Go back and try again.
• A little review will surely help.
• Click to Review Try Again!

81
Real life Applications
• Trigonometry is used to solve real life problems.
• The following slides show a few examples where
trigonometry is used.
• Search the Internet for more examples if you
like.

CLICK TO CONTINUE
82
Real life example 1
• Measuring the height of trees

What would you need to know in order to calculate
the height of this tree? What trig ratio would
you use?
CLICK TO CONTINUE
83
Finding height of towers..
• Tall buildings (skyscrapers), towers and
mountains

CLICK TO CONTINUE
84
Solve real problem
• Assume the line in the middle of the drawn
triangle is perpendicular to the beach line.
• How far is the island from the beach?

yours
CLICK TO CONTINUE
85
Distance to the beach(Ans.)
• The distance we want is the shortest distance
?.
• The tangent ratio can be used here
• tan 30o x/50
• 0.57735 x/50
• x 0.57735(50) 28.8675
• Therefore the island is about 29m from the beach.

86
• h is the height of the tree. That is what we are
looking for.
• We need to know the angle of elevation ? and
also the horizontal distance from A to the bale
od the tree, x
• The tangent ratio would be used
• tan ? h/x and so h xtan ?

GO BACK
87
Congratulations
• Be proud of yourself. You have successfully
completed a crash course in basic trigonometry
and I expect you to be able to do well on this
strand in the Common Core States Standards test.
Print the certificate to show your achievement.

CLICK TO CONTINUE
88
Certificate of Completion
• I hereby certify that
• _____________________________________
• has satisfactorily completed a basic course in
Introduction to Trigonometry on this day the
____________________ of the year 20___
• The bearer is qualified to solve some real world
problems using trigonometry.
• Signed Nevermind E. Chigoba