Chapter 6.1 – 6.3

An Introduction to Trigonometry

1Page 36

Note

• This presentation contains material not explicitly given in

your textbook

2

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

3

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

4

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

• …

6

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

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?

• 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

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

• How long is the 3rd side?

Sides are x, 3x and 2x

2x

x

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?

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=a, B=b, C=c, AB/AC= ab/ac, etc.

A

C B

a

bc

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

Scout

Tree Shadow

Scout

Shadow

Tree Height/Tree Shadow =

Scout Height/Scout Shadow

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

• 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!

y = m x

Is the

measure of

the angle

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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.

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

cotangent = x/y

For x, y non-zero

Alternate Approach

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x

y

Opposite

Adjacent

Hypotenuse

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

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!

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x

y

Opposite

Adjacent

Hypotenuse

(x, y)

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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?

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 1 All > 0Q 2 Sin >0

Q 3 Tan >0Q 4 Cos >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 ?

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:

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