1 Right Triangle Trigonometry Pre-Calculus Monday, April 20.

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Right Triangle Trigonometry

Pre-Calculus

Monday, April 20

Today’s Objective

Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions

Begin learning some of the Trigonometric identities

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• Evaluate trigonometric functions of acute angles.

• Use fundamental trigonometric identities.

• Use a calculator to evaluate trigonometricfunctions.

• Use trigonometric functions to model and solvereal-life problems.

What You Should Learn

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Plan Questions from last week? Notes! Guided Practice Homework

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Right Triangle Trigonometry

Trigonometry is based upon ratios of the sides of right triangles.

The ratio of sides in triangles with the same angles is consistent. The size of the triangle does not matter because the triangles are similar (same shape different size).

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The six trigonometric functions of a right triangle,

with an acute angle , are defined by ratios of two

sides of the triangle.

The sides of the right triangle are:

the side opposite the acute angle

the side adjacent to the acute angle ,

and the hypotenuse of the right triangle.

opp

adj

hyp

θ

θ

θ

θ

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The trigonometric functions are

sine, cosine, tangent, cotangent, secant, and cosecant.

opp

adj

hyp

θ

Sin = cos = tan =

Csc = sec = cot =

opphyp

adj

hyp

hypadj

adj

opp

oppadj

hyp

opp

Note: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals.

θ

θ

θ

θ

θ

θ

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Reciprocal Functions

Another way to look at it…

sin = 1/csc csc = 1/sin

cos = 1/sec sec = 1/cos tan = 1/cot cot = 1/tan

Given 2 sides of a right triangle you should be able to find the value of all 6 trigonometric functions.

Example:

9

5

12

10

Calculate the trigonometric functions for .

The six trig ratios are 4

3

5

Sin =5

4

Tan =3

4

Sec =3

5

Cos =5

3

Cot =4

3

Csc =4

5

cos α =5

4

sin α =5

3

cot α =3

4

tan α =4

3

csc α =3

5

sec α =4

5

What is the relationship of

α and θ?

They are complementary (α = 90 – θ)

Calculate the trigonometric functions for .

θ

θ

θ

θ

θ

θ

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Note : These functions of the complements are called cofunctions.

Note sin = cos(90 ), for 0 < < 90

Note that and 90 are complementary angles.

Side a is opposite θ and also adjacent to 90○– θ .

ahyp

bθ

90○– θ

sin = and cos (90 ) = .

So, sin = cos (90 ).

b

a

b

a

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Cofunctions

sin = cos (90 ) cos = sin (90 )

sin = cos (π/2 ) cos = sin (π/2 )

tan = cot (90 ) cot = tan (90 )

tan = cot (π/2 ) cot = tan (π/2 )

sec = csc (90 ) csc = sec (90 )

sec = csc (π/2 ) csc = sec (π/2 )

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Trigonometric Identities are trigonometric equations that hold for all values of the variables.

We will learn many Trigonometric Identities and use them to simplify and solve problems.

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Quotient Identities

Sin = cos = tan = hyp

adjadj

opp

hyp

opp

opp

adj

hyp

θ

tancos

sin

adj

opp

adj

hyp

hyp

opp

hypadjhypopp

The same argument can be made for cot… since it is the reciprocal function of tan.

θ θθ

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Quotient Identities

cos

sintan

sin

coscot

Pythagorean Identities

Three additional identities that we will use are those related to the Pythagorean Theorem:

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sin2 + cos2 = 1

tan2 + 1 = sec2

cot2 + 1 = csc2

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Some old geometry favorites…

Let’s look at the trigonometric functions of a few familiar triangles…

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Geometry of the 45-45-90 triangle

Consider an isosceles right triangle with two sides of length 1.

1

1

45

452211 22

The Pythagorean Theorem implies that the hypotenuse

is of length .2

Remember a2 + b2 = c2

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Calculate the trigonometric functions for a 45° angle.

2

1

1

45

csc 45° = = =

1

2 2opphypsec 45° = = =

1

2 2adj

hyp

cos 45° = = =

2

2

2

1

hypadjsin 45° = = =

2

2

2

1

hyp

opp

cot 45° = = = 1

oppadj

1

1tan 45° = = = 1 adj

opp1

1

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60○ 60○

Consider an equilateral triangle with each side of length 2.

The perpendicular bisector of the base bisects the opposite angle.

The three sides are equal, so the angles are equal; each is 60°.

Geometry of the 30-60-90 triangle

2 2

21 1

30○ 30○

3

Use the Pythagorean Theorem to find the length of the altitude, . 3

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Calculate the trigonometric functions for a 30 angle.

12

30

3

csc 30° = = = 2 1

2opphyp

sec 30° = = =

3

2

3

32

adj

hyp

cos 30° = = 2

3

hypadj

tan 30° = = =

3

1

3

3

adj

oppcot 30° = = =

1

3 3oppadj

sin 30° = =

2

1

hyp

opp

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Calculate the trigonometric functions for a 60 angle.

1

2

60○

3

csc 60 = = =

3

2

3

32opphyp

sec 60 = = = 2

1

2

adj

hyp

cos 60 = = 2

1

hypadj

tan 60 = = =

1

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adj

opp cot 60 = = = 3

1

3

3

oppadj

sin 60 = = 2

3

hyp

opp

Some basic trig values

Sine Cosine Tangent

300

/6

450

/41

600

/3

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2

3

2

3

2

1

2

1 3

3

3

2

2

2

2

IDENTITIES WE HAVE REVIEWED SO FAR…

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Fundamental Trigonometric Identities

Co function Identitiessin = cos(90 ) cos = sin(90 )sin = cos (π/2 ) cos = sin (π/2 )tan = cot(90 ) cot = tan(90 )tan = cot (π/2 ) cot = tan (π/2 )sec = csc(90 ) csc = sec(90 ) sec = csc (π/2 ) csc = sec (π/2 )

Reciprocal Identities

sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin

Quotient Identities

tan = sin /cos cot = cos /sin


sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2

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Example: Given sec = 4, find the values of the other five trigonometric functions of .

Use the Pythagorean Theorem to solve for the third side of the triangle.

tan = = cot =1

1515

115

sin = csc = =4

15

15

4

sin

1

cos = sec = = 4 4

1cos

1

15

θ

4

1

Draw a right triangle with an angle such that 4 = sec = = .

adjhyp

1

4

Using the calculator

Function Keys

Reciprocal Key

Inverse Keys

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Using Trigonometry to Solve a Right Triangle

A surveyor is standing 115 feet from the base of the

Washington Monument. The surveyor measures the

angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument?

Figure 4.33

Applications Involving Right Triangles

The angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object.

For objects that lie below the horizontal, it is common to use the term angle of depression.

Solution

where x = 115 and y is the height of the monument. So, the height of the Washington Monument is

y = x tan 78.3

115(4.82882) 555 feet.

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Homework4-2 Practice 1

1-17 ODD

Tuesday, April 21, 2015

BENCHMARK TOMORROW

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Kahoot all day long!

Here’s the deal…1. You need a PENCIL and PAPER

2. You will be GRADED for participating so make sure your Kahoot name is your real name

3. If you get kicked out, you must log back in

4. If you do not have an ipad/smart phone, you may work in teams of TWO or do your work on a piece of paper and turn that in

5. THIS IS REQUIRED.

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Review Exponents and Log

Converting Logs and Exponents https://play.kahoot.it/#/k/68a4661b-b90b-498

7-820e-956c7e5af6bd

Log and Inverses https://play.kahoot.it/#/k/45a02d91-aa36-485f

-ba52-436768d981f7

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Review Intro to Trig

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Right Triangle Trig and Angle Measures https://play.kahoot.it/#/k/8858d7ed-b092-46c6

-bc5d-b16c044665c0

Radians, Degrees, Arc Length https://play.kahoot.it/#/k/63ac7b20-59ce-4fd4-

9266-954797b1b39a

Wednesday

BENCHMARK!

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Thursday, April 23

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Do Now – How was the benchmark? What can you improve?

Benchmark Data

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Benchmark Data 1st Period

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Analysis By Question

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44


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Benchmark Data 2nd Period

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48


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50


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Special Angle Names

Angle of ElevationFrom Horizontal Up

Angle of DepressionFrom Horizontal Down

Angle of Elevation and Depression

The angle of elevation is measured from the horizontal up to the object.

Imagine you are standing here.

Angle of Elevation and Depression

The angle of depression is measured from the horizontal down to the object.

Constructing a right triangle, we are able to use trig to solve the triangle.

Guided Practice! Follow along on your handout!

Lighthouse & SailboatSuppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat?

Construct a triangle and label the known parts. Use a variable for the unknown value.

5.7o

150 ft.

x

150 ft.

Lighthouse & SailboatSuppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat?

5.7ox

Set up an equation and solve.

150 ft.

Lighthouse & Sailboat

150tan(5.7 )o

x

tan(5.7 ) 150ox

150

tan(5.7 )ox

Remember to use degree mode!

x is approximately 1,503 ft.

5.7o

x

150 ft.

River Width

A surveyor is measuring a river’s width. He uses a tree and a big rock that are on the edge of the river on opposite sides. After turning through an angle of 90° at the big rock, he walks 100 meters away to his tent. He finds the angle from his walking path to the tree on the opposite side to be 25°. What is the width of the river?

Draw a diagram to describe this situation. Label the variable(s)

River Width

tan(25 )100

d

We are looking at the “opposite” and the “adjacent” from the given angle, so we will use tangent

Multiply by 100 on both sides

100 tan(25 ) d

46.63 metersd

Subway The DuPont Circle Metrorail Station

in Washington DC has an escalator which carries passengers from the underground tunnel to the street above. If the angle of elevation of the escalator is 52° and a passenger rides the escalator for 188 ft, find the vertical distance between the tunnel and the street. In other words, how far below street level is the tunnel?

Subway

We are looking at the “opposite” and the “hypotenuse” from the given angle so we will use sine

sin(52 )188

h

188sin(52 ) h

Multiply by 188 on each side

148.15 feeth

Building HeightA spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35o. The angle of elevation to the top of the spire is 38o. How tall is the spire?

Construct the required triangles and label.

500 ft.

38o 35o

Building HeightWrite an equation and solve.

Total height (t) = building height (b) + spire height (s)

500 ft.

38o 35o

Solve for the spire height.

t

b

s

Total Height

tan(38 )500

o t

500 tan(38 )o t

Building HeightWrite an equation and solve.

500 ft.

38o 35o

Building Height

tan(35 )500

o b

500 tan(35 )o b t

b

s

Building Height

5050 0 t0 tan an(3(38 ) 5 )o o s

Write an equation and solve.

500 ft.

38o 35o

500 tan(38 )o t 500 tan(35 )o b

5050 0 t0 tan an(3(38 ) 5 )o o s

The height of the spire is approximately 41 feet.

t

b

s

Total height (t) = building height (b) + spire height (s)

Mountain Height

A hiker measures the angle of elevation to a mountain peak in the distance at 28o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29o. How much higher is the mountain peak than the hiker?

Construct a diagram and label.

1st measurement 28o.

2nd measurement 1,500 ft closer is 29o.

Mountain Height

Adding labels to the diagram, we need to find h.

28o 29o

1500 ft x ft

h ft

Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28o is 1500 + x.

tan 281500

o h

x

tan 29o h

x

Mountain Height

tan 29o h

x

tan(29 )ox h

Now we have two equations with two variables.Solve by substitution.

tan 281500

o h

x

(1500 ) tan(28 )ox h

(1500 ) tan(28 ) tan(29 )o ox x

Solve each equation for h.

Substitute.

Mountain Height

1500 tan(28 ) tan(28 ) tan(29 )o o ox x

(1500 ) tan(28 ) tan(29 )o ox x Solve for x. Distribute.

Get the x’s on one side and factor out the x.

Divide.

1500 tan(28 ) tan(29 ) tan(28 )o o ox x

1500 tan(28 ) tan(29 ) tan(28 )o o ox

1500 tan(28 )

tan(29 ) tan(28 )

o

o ox

Mountain Height

tan(29 )ox h

However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height.

1500 tan 28 tan 2919,562

tan 29 tan 28

o

o

o o

The height of the mountain above the hiker is 19,562 ft.

1 Right Triangle Trigonometry Pre-Calculus Monday, April 20.

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