Transcript Introduction to Trigonometry: sines, cosines, tangents

Introduction to Trigonometric
Functions
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• Trig functions are the
relationships amongst
various sides in right
triangles.
• You know by the
Pythagorean theorem
that the sum of the
squares of each of
the smaller sides
equals the square of
the hypotenuse, a2  b2  c2
You know in the above triangle that
2
2
2
a b  c
Trig functions are how the relationships
amongst the lengths of the sides of a
right triangle vary as the other angles
are changed.
How does this relate to trig?
• The opposite side divided by the
hypotenuse, a/c, is called the sine of angle
A
• The adjacent side divided by the
hypotenuse, b/c, is called the cosine of
Angle A
• The opposite side divided by the adjacent
side, a/b, is called the tangent of Angle A
Remember SOHCAHTOA
• Sine is Opposite
divided by
Hypotenuse
• Cosine is Adjacent
divided by
Hypotenuse
• Tangent is Opposite
divided by Adjacent
• SOHCAHTOA!!!!!!
Table of Contents
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Examples
Question 1
Question 2
Question 3
Question 4
Example 1
If a = 3 and c = 6, what is the
measurement of angle A?
Answer: a/c is a sine relationship
with A.
Sine A = 3/6 = .5, from your
calculator, angle A = 30 degrees.
Example 2
• A flagpole casts a 100
foot shadow at noon.
Lying on the ground
at the end of the
shadow you measure
an angle of 25
degrees to the top of
the flagpole.
• How High is the
flagpole?
How do you solve this question?
• You have an angle, 25
degrees, and the length of the
side next to the angle, 100
feet. You are trying to find the
length of the side opposite the
angle.
• Opposite/adjacent is a tangent
relationship
• Let x be the height of the
flagpole
• From your calculator, the
tangent of 25 is .47
x
• .47 = 100
• x = (.47)(100), x = 47
• The flagpole is 47 feet high.
Question 1
• Given Angle A is 35 degrees, and b = 50
feet.
• Find c. Click on the correct answer.
• A. 61 feet
• B 87 feet
• C. 71 feet
GREAT JOB!
• You have an angle and an
adjacent side, you need to
find the hypotenuse. You
knew that the cosine finds
the relationship between the
adjacent and the
hypotenuse.
• Cosine 35 = 50/c, c Cosine
35 = 50,
• So c = 50/cos 35, or
approximately 61
Next question
Nice try
• You have an angle
and the adjacent side.
You want to find the
hypotenuse.
• What relationship
uses the adjacent and
the hypotenuse?
Back to
Question
Back to tutorial
Question 2
• If the adjacent side is 50, and the
hypotenuse is 100, what is the angle?
Please click on the correct answer.
• A. 60 degrees
• B. 30 degrees
• C. 26 degrees
Way to go!
• Given the adjacent
side and the
hypotenuse, you
recognized that the
adjacent divided by
the hypotenuse was a
cosine relationship.
• Cosine A = 50/100,
• A = 60 degrees
Next question
Nice try
• Given an adjacent
side and a
hypotenuse, what
relationship will give
you the angle?
Back to question
Back to tutorial
Question 3
• If the opposite side is 75, and the angle is
80 degrees, how long is the adjacent side?
• A. 431
• B. 76
• C. 13
Nice job
• You were given the
opposite side of 75 and
an angle of 80 degrees
and were asked to find
the adjacent side. You
recognized that this was
a tangent relationship.
• Tangent 80 = 75/b,
• b tangent 80 = 75,
• b=
= 13
75
tan 80
Next question
Nice Try
• You are given an
angle and the
opposite side, and
have been asked to
find the adjacent side.
What relationship
uses the opposite
side and the adjacent
side?
Back to question
Back to tutorial
Question 4:
If B = 50 degrees and b = 100
what is c?
B
B. 130
C. 84
c
________
A. 155
a
A ________ C
b
Nice try
• What is the
relationship between
B and b? And, what is
the relationship
between b and c?
Return to question
Return to tutorial
Great job!
• First, you recognized that
b is the opposite side
from B. Then, you
recognized that the
relationship between an
opposite side and the
hypotenuse is a sine
relationship.
• Sine 50 = 100/c, c Sine
50 = 100, c = 100/sine 50
= 130.
Go to next section
Introduction to Quadrants
II
__________________
90 degrees
I
180 degrees______________________ 0 degrees
III
270 degrees
IV
Quadrants
• All angles are divided into 4 quadrants
• Angles between 0 and 90 degrees are in
quadrant 1
• Angles between 90 and 180 degrees are in
quadrant II
• Angles between 180 and 270 degrees are in
quadrant III
• Angles between 270 and 360 degrees are in
quadrant IV
• Why is this important? Click and find out!
Importance of quadrants
• Different trig functions are positive and
negative in different quadrants.
• The easy way to remember which are
positive and negative in each quadrant it
to remember, “All Students Take Classes”
All Students Take Classes
• Quadrant I: 0 – 90 degrees: All: All trig
functions are positive
• Quadrant II: 90 – 180 degrees: Students:
Sine functions are positive
• Quadrant III: 180 – 270 degrees: Take:
Tangent functions are positive
• Quadrant IV: 270 – 360 degrees: Classes;
Cosine functions are positive
Standard Angle Values
Angle
Sine
Cosine
Tangent
0
30
45
60
90
0 1 2 3 4
2 2 2 2 2
4
2
3
2
2
2
1
2
0
2
0
4
1
3
2
2
3
1
4
0
Remember
• Simplify the fractions
• Place the radicals in
the numerator. Write
• Instead of 1
2
2
2
Congratulations
• You have learned how to use the 3 main
trig functions, you have learned which
functions are positive in which quadrants,
and you have learned values of sine,
cosine, and tangent for 5 standard angles.
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