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Chapter 6.1 – 6.3
An Introduction to Trigonometry
Page 36
1
Note
• This presentation contains material not explicitly given in
your textbook
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History of Trigonometry
• The subject name is from the Greeks
– Trigonon: Triangle
– Metron: Measure
• The Greeks developed the subject largely as a form of
measurement for Astronomy ~ 3 BC
• As the name indicates, it is largely the study of triangles, or
perhaps the study of geometry using triangles
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Euclid
• The Greek Alexandrian, Euclid, is by far the name most
commonly associated with Trigonometry
• “Author” of a thirteen volume treatise
The Elements
– Most commonly used textbook until the 20th century
(only the Bible has been published in more editions)
– Every “educated” person used it, and every intellectual had
a copy on his bookshelf
– The book is the oldest deductive mathematics text
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Our Class
• We are going to use deductive mathematics to explore a
small part of Euclid’s world
• Our study of trigonometry is restricted to
– Planes
– Right triangles
(Confining our study to right triangles is not restrictive,
since any triangle can be divided into two right triangles)
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Applications of Trigonometry
• Triangulation: Measuring distances/locations
– Used in satellite navigation
– Geography
– Astronomy
• Studies of periodic waves, such as sound and light:
– Acoustics
– Medical imaging
– Optics
• Measurement
– Surveying
– Civil Engineering
• Animation
• …
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Some Basics
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Angle Vocabulary
•
Basics:
– Ray – a half line
– Angle – joining of two rays
– Vertex – common endpoint of rays forming an angle
•
Special Angles:
– Straight angle – 180
– Right angle – 90 
– Complementary angles – sum to 90 
– Supplementary angles – sum to 180 
•
Decimals vs. Degrees/Minutes/Seconds
– 60 min in a degree
– 60 sec in a minute
– Converting
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Coterminal rays
• Coterminal means that the ends of the ray (vertex) are
the same
• Measure of an angle is from initial to terminal side (ray)
– 0 is along the x axis
– Positive angles are counterclockwise
– Negative angles are clockwise
– 360  is a full revolution
• We commonly use Greek letters for angles, e.g., 
y

x
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Why 360???
• Around 1500 BC, Egyptians divided the day into 24 hours,
though the hours varied with the seasons originally
• Greek astronomers made the hours equal.
• About 300 to 100 BC, the Babylonians subdivided the hour
into base-60 fractions: 60 minutes in an hour and 60 seconds
in a minute.
• Perhaps 360 comes from an estimate of the number of days
in a year?
• Additionally, 360 has lots of factors
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What are the degree angles of the four quadrants?
• Q1?
• Q2?
• Q3?
• Q4?
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• Q1? 0 to 90
• Q2? 90 to 180
• Q3? 180 to 170
• Q4? 170 to 360
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Special Triangles
• Right Triangle: two sides form 90  angle
– Longest side is the hypotenuse
– Other two sides are legs
• 45-45-90 triangle
– Refers to the angles of the triangle
– Since Pythagorean Theorem says:
a2 +b2 = c2, we can easily find the length of the sides:
c2 = 2x2 , c = x 2
The sides of a 45-45-90 triangle are:
x, x, x 2
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Question
• How large are the angles of an equilateral triangle??
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Examples
• The leg of a right triangle that has a 45 degree angle has
length 10 inches. What is the length of the rest of the sides?
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Solution
• We know that the two legs are the same length. If the length
of the legs is x, then the hypotenuse is 2 x.
• The hypotenuse is 10 2
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30-60-90 Triangle
• Take an equilateral triangle and cut it in half
2x
x
• How long is the 3rd side?
Sides are x, 3x and 2x
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Example
• The shortest side of a 30-60-90 triangle is 2 yards. How long
are the rest of the sides? What is its area?
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Solution
• The shortest side of a 30-60-90 triangle is 2 yards. How long
are the rest of the sides? What is its area?
• Sides are x,
3x and 2x, so we have 2, 2 3 and 4 as the
sides.
• The area is the base x height. Here, those are the legs.
The area is 3 x2
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Summary
• Vocabulary:
– Ray, Angle, Vertex
– Right, Straight, Complementary, Supplementary Angles
• Angles:
– Degrees, Minutes
– Coterminal
– Positive, Negative
• Triangles:
– Right
– 45-45-90
– 30-60-90
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Triangle Types
• Right triangle – one side is 90
• Acute – all three angles < 90 
• Obtuse – one angle > 90 
• Equilateral – all sides equal
• Isosceles – two sides equal
• Scalene – no sides equal
• Questions:
– Can an isosceles triangle be a right triangle?
– Can an equilateral triangle be a right triangle?
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Solution
• Questions:
– Can an isosceles triangle be a right triangle? yes
– Can an equilateral triangle be a right triangle? no
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Properties of Triangles
• The sum of all the angles of a triangle is 180 
• The sum of any two sides is larger than the third
• The largest angle is opposite the largest side;
similarly, the smallest angle is opposite the smallest side
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Similar Triangles
• In similar triangles, corresponding angles are equal
• In similar triangles, corresponding sides are proportional
A
C
B
a
b
c
• A=a, B=b, C=c, AB/AC= ab/ac, etc.
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For Similar Triangles
• Need one of the three:
Two angles the same (implies all three)
Two sides proportional and the angle between them equal
Three sides proportional
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How do we use similar triangles?
• Classic problem, how tall is a tree? (Boy Scout Problem)
– Measure the length of the tree’s shadow
– Measure the length of a shadow of something known
Tree Height/Tree Shadow =
Scout Height/Scout Shadow
Tree
Scout
Scout
Shadow
Tree Shadow
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Example
• We have two triangles ABC and DEF. Angle A = Angle D and
Angle C = Angle F.
• Side b = 4 and side e = 6. If side a is 8, how long is side d?
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Solution
• We have two triangles ABC and DEF. Angle A = Angle D and
Angle C = Angle F.
• Side b = 4 and side e = 6. If side a is 8, how long is side d?
• Since two angles are equal, all three are and the triangle are
similar
• Since b/e = 4/6 = 2/3, we have a/d = 2/3 = 8/12. Side d is 12
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Summary
• Types of triangles
– Obtuse, Acute
– Equilateral, Isosceles, Scalene
• Largest side is opposite largest angle, etc.
• Similar triangles
– Equal angles
– Proportional sides
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Standard Position
• An angle is in Standard Position if its vertex is at the origin of
the axes and its initial side is along the x axis
Angle
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The Equation of the Coincident Line
• Slope, m = (y2 – y1)/(x2 – x1)
• But one of the points is (0,0), so the slope is y/x, where (x,y) is any
point on the line
 Is the
measure of
the angle

y=mx
• So, given any point on the line formed by the ray, we have the slope
• Note, every triangle formed by the line and the x axis is similar!
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Example
• For an angle in standard form with one ray going through the
point (2,3) what is the equation of the line?
• Give two other points that are on the line.
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Solution
• For an angle in standard form with one ray going through the
point (2,3) what is the equation of the line?
y = 3/2 x
• Give two other points that are on the line.
(4,6), (8, 12)
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More Characteristics
• Consider the length of the ray, r formed by the angle 
– r is a side of a triangle
– All of the triangles formed by the ray are similar, since they
have the same angles
• Because similar triangles have proportional sides we know
that x/r and y/r are constant for any point on the ray!
• If x/r is constant, so is r/x and, therefore, r/y
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Key Point
For every point on the ray, we have:
x/y, y/x, x/r constant and r/x, y/r, r/y constant!
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Definitions
In Trigonometry, we give names to these ratios:
sine  = y/r
cosine  = x/r
tangent  = y/x
cosecant  = r/y
secant  = r/x
For x, y non-zero
cotangent  = x/y
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Alternate Approach
y
Hypotenuse
Opposite

Adjacent
x
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite /Adjacent
SOHCAHTOA
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Abbreviations
• Cosine: cos
• Sine: sin
• Tangent: tan
• Cosecant: csc
• Secant: sec
• Cotangent: cot
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Example
• If a terminal side of an angle  is the line containing the
point (3,4), what are the values of the trigonometric functions?
– sin 
– cos 
– tan 
– csc 
– sec 
– cot 
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Solution
• If a terminal side of an angle  is the line containing the
point (3,4), what are the values of the trigonometric functions?
Need r = sqrt (9 + 16) = sqrt ( 25 ) = 5
– sin  = 4/5
– cos  = 3/5
– tan  = 4/3
– csc  = 5/3
– sec  = 5/4
– cot  = 3/4
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Slope vs. Tangent
• Slope is defined as the vertical change/ horizontal change
• If one point of the line is (0, 0), the origin, the slope is the
y value of the point / the x value
• Therefore, the slope is just the tangent!
y
(x, y)
Hypotenuse
Opposite

Adjacent
x
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More Examples
• The terminal side of an angle is in the second quadrant (QII)
coincident with the line y = - 12/5 x. Find sin, cos, tan
• Suppose the terminal side of the angle is in QIV? What are
sin, cos, tan?
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Solutions
• The terminal side of an angle is in the second quadrant (QII)
coincident with the line y = - 12/5 x
sin  = 12/13
cos  = - 5/13
tan  = -12/5
• Suppose the terminal side of the angle is in QIV?
the signs of the cos function is + and the sin -
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Example
• Suppose a ray starting at the origin goes through the point
(2,3) in the plane, making an angle with the origin of A
• What are sin A, cos A and tan A?
• What is the equation of the line coincident with the ray?
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Solution
• Suppose a ray starting at the origin goes through the point
(2,3) in the plane, making an angle with the origin of A
• If the point is (2,3), we have the x distance 2, and y dist 3
need the hypotenuse: 4 + 9 = 13, h = 13
• Cos A = 2/ 13 , sin A = 3/ 13 , tan = 3/2
• Tan also gives the slope
• y = 2/3x + b, b = 0 because goes through the origin
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Reciprocal Identities
• sin a = 1/csc a
• cos a = 1/sec a
• tan a = 1/cot a
• csc a = 1/sin a
• sec a = 1/cos a
• cot a = 1/tan a
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Signs of Trig Functions
Q 2 Sin >0
Q 1 All > 0
Q 4 Cos >0
Q 3 Tan >0
ASTC – All Students Take Classes
Reciprocal functions have the same sign
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Examples: Which Quadrant?
• Sin > 0, Tan < 0 ?
• Cos > 0, Tan < 0 ?
• Sin > 0, Tan > 0 ?
• Cos > 0, Tan > 0?
• Sin > 0, Cos > 0?
• Csc > 0?
• Cot > 0 ?
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Solution
• Sin > 0, Tan < 0 ? 2
• Cos > 0, Tan < 0 ? 4
• Sin > 0, Tan > 0 ? 1
• Cos > 0, Tan > 0? 1
• Sin > 0, Cos > 0? 1
• Csc > 0? 3, 4
• Cot > 0 ? 1, 3
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What happens on the axes?
• What are the trig functions at 0, 90, 180, and 270 deg?
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Values of Trig Functions on Axes
0
90
180
270
Sin
0
1
0
-1
Cos
1
0
-1
0
Tan
0
undef
0
undef
Sec
1
undef
-1
undef
Csc
undef
1
undef
-1
Cot
undef
0
undef
0
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Examples: Value of Functions
• If cos  = -5/13 and sin  > 0, find the values
of other functions:
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Solution
• If cos  = -5/13 and sin  > 0, find the values
of other functions:
Sin  = 12/13
Tan  = -12/5
Csc  = -13/5
Sec  = -5/12
Cot  = -5/12
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Note
• We will use degrees for angles unless otherwise stated!
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Trig Values of 45-45-90 Triangles
• Sides are x, x, x 2
• Which is the largest side?
• Sin 45 = 1/ 2
• Cos 45 = 1/ 2
• Tan 45 = 1
• Find csc, sec, cot for 45 deg
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Solution
• Find csc, sec, cot for 45 deg
• Csc - 2
• Sec= 2
• Cot = 1
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Trig Values of 30-60-90 Triangles
• Sides are x, 2x, x 3
• Which side is smallest, largest?
• cos 30 = 3/2
• sin 30 = 1/2
• tan 30 = 1/ 3 = 3/3
• cos 60 = 1/2
• sin 60 = 3/2
• tan 60 =
3
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Example
• Suppose a is an acute angle and cos a = 2/5
• Find sin a and tan a
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Solution
• Suppose a is an acute angle and cos a = 2/5
• Find sin a and tan a
• Use Pythagorean Theorem
4 + x2 = 25, x = 21
• Sin a = 21/5
• Tan a = 21/2
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What is the angle?
• Cos A = 0
• Sin B = 1
• Tan C = 1
• Cot D = -1
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Solution
• Cos A = 0 A = 0, 180
• Sin B = 1 B = 90
• Tan C = 1 C = 45, 225
• Cot D = -1 D = 135, 315
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Problems
Using the values given, find the six trig functions in Quadrant I
• sec a = 6/5
• tan b = 1/2
• sin d = 3/4
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Solutions
Using the values given, find the six trig functions in Quadrant I
• sec a = 6/5, a2 + 25 = 36, a = 11
cos= 5/6, sin a = 11/6, tan = 11/5, cot = 5/ 11, csc= 6/ 11
• tan b = ½, 1 + 4 = 5, b = 5
Cot = 2, sin=1/ 5, cos = 2/ 5, csc = 5, sec = 5/2
• sin d = ¾, 9 + d2 = 16, d= 5
Cos = 5/4, tan = 3/ 5, sec = 4/ 5, csc = 4/3, cot = 5/3
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Example
Find the sides of a triangle with
• tan b = -1/2
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Find sides of a triangle with
• tan b = -1/2
• Needs to be in quadrant 2 or 3
• Sides are 1, 2, 5 or multiples of them
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Summary
• Trigonometric Functions Defined
• Understand the signs of the angles in the coordinate plane
• Can evaluate trig functions in quadrants and on axes
• Reciprocal Identities
• 30-60-90 and 45-45-90 trig values
• Using Pythagorean Theorem
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