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Trigonometry and VectorsCommon triangles in Geometry and

Trigonometry

11

1

2

45o

45o

2

3

30o

60o

You must memorize these triangles

2 3

Trigonometry and Vectors

Common triangles in Geometry and Trigonometry

3

4

5

1

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POLY ENGINEERING3-8


Trigonometric functions are ratios of the lengths of the segments that make up angles.

Trigonometric Functions

tan A = opposite adjacent

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

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POLY ENGINEERING3-9


1. Scalar Quantities – a quantity that involves magnitude only; direction is not importantTiger Woods – 6’1”Shaquille O’Neill – 7’0”

2. Vector Quantities – a quantity that involves both magnitude and direction

Vectors

How hard to impact the cue ball is only part of the game – you need to know direction too

Weight is a vector quantity

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POLY ENGINEERING3-9


1. 5 miles northeast

2. 6 yards

3. 1000 lbs force

Scalar or Vector?

VectorMagnitude and Direction

ScalarMagnitude only


4. 400 mph due north

5. $100

6. 10 lbs weight




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POLY ENGINEERING3-9


3. Free-body DiagramA diagram that shows all external forces acting on an object.

Vectors

friction force

force of gravity

(weight)

applied force

normal force

Wt

FN

Ff

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POLY ENGINEERING3-9


4. Describing vectors – We MUST represent both magnitude and direction.

Describe the force applied to the wagon by the skeleton:

Vectors

45o40 lb

s

magnitude direction

F = 40 lbs 45o

Hat signifies vector quantity

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POLY ENGINEERING3-10


1. We can multiply any vector by a real number.2. Original direction is maintained, new magnitude.

Vectors – Scalar Multiplication

2

½

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1. We can add two or more vectors together. 2. Redraw vectors head-to-tail, then draw the resultant vector.

(head-to-tail order does not matter)

Vectors – Addition

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Find 2 a

March 14, 2011Drill

y

x

ay

x

a

b

Find a + b

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Find 2 a

Drill

y

x

a

2a

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Drill

y

x

a

b

Find a + b

aa+b

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Drill

y

x

a

b

Find a + b

a+b

b

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Trigonometry and VectorsVectors – Rectangular Components

y

x

F

Fx

Fy

1. It is often useful to break a vector into horizontal and vertical components (rectangular components).

2. Consider the Force vector below. 3. Plot this vector on x-y axis.4. Project the vector onto x and y axis.

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Trigonometry and VectorsVectors – Rectangular Components

y

x

F

Fx

Fy

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:

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Vectors – Rectangular Components

y

x

F

Fx

Fy

Fx = Fx i

Vector Fx = Magnitude Fx times vector i

Vector Fy = Magnitude Fy times vector j

Fy = Fy j

F = Fx i + Fy j

i denotes vector in x direction

j denotes vector in y direction

Unit vector

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From now on, vectors on this screen will appear as bold type without hats.

For example, Fx = (4 lbs)i

Fy = (3 lbs)j

F = (4 lbs)i + (3 lbs)j

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y

x

F

Fx

Fy

Each grid space represents 1 lb force.

What is Fx?

Fx = (4 lbs)i

What is Fy?

Fy = (3 lbs)j

What is F?

F = (4 lbs)i + (3 lbs)j

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F

Fx

Fy

cos = Fx / F

Fx = F cos i

sin = Fy / F

Fy = F sin j

What is the relationship between , sin , and cos ?

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y

x

F Fx +

Fy +

When are Fx and Fy Positive/Negative?

FFx -

Fy +

FFFx -Fy -

Fx +Fy -

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Complete the following chart in your notebook:

III

III IV

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y

x

F

Each grid space represents 1 lb force.

What is Fx?

Fx = (-1 lbs)i

What is Fy?

Fy = (3 lbs)j

What is F?

F = (-1 lbs)i + (3 lbs)j

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Trigonometry and VectorsVectors – Resultant Forces

Resultant forces are the overall combination of all forces acting on a body. 1) Break up all forces into x and y component forces

2) add up all of the component forces in x-direction

3) add up all of the component forces in y-direction

4) Write resultant as single vector in rectangular components

100 lb

150 lb

60o

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Trigonometry and Vectors1) Break up all forces into x and y component forces

Space Junk:

150 lb

60o

iCosFx )60(*150

iiFx 752

1*150

jSinFy )60(*150

jFy 375

jiF 37575

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Trigonometry and VectorsBreak up all forces into x and y component forces

Gravity

100 lb

iFx 0

jFy 100

jF 100

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Trigonometry and Vectors2) Add up all forces in x direction

100 lb

150 lb

60o

iFx 75

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Trigonometry and Vectors3) Add up all forces in y direction

100 lb

150 lb

60o

jjFy 100375

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Trigonometry and Vectors4) Write resultant as single vector in

rectangular components

100 lb

150 lb

60o

jiF 10037575

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Classwork

Complete Worksheet

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Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction

2) sum of forces in y-direction

3) Write as single vector in rectangular components

Fx = F cos Qi

= (150 lbs) (cos 60) i

= (75 lbs)i

SFx = (75 lbs)i

No x-component

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Fy = F sin Qj

= (150 lbs) (sin 60) j

= (75 lbs)j

Wy = -(100 lbs)j

SFy = (75 lbs)j - (100 lbs)j

SFy = (75 - 100 lbs)j

3

3

3

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R = SFx + SFy

R = (75 lbs)i + (75 - 100 lbs)j

R = (75 lbs)i + (29.9 lbs)j

3





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CLASSWORK / HOMEWORK

Complete problem #4 on the Vector Worksheet

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