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Trigonometry and VectorsCommon triangles in Geometry and

Trigonometry

You must memorize these triangles

Trigonometry and Vectors

Common triangles in Geometry and Trigonometry

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

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

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

ScalarMagnitude only

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

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

magnitude direction

F = 40 lbs 45o

Hat signifies vector quantity

POLY ENGINEERING3-10

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

Vectors – Scalar Multiplication

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

Find 2 a

March 14, 2011Drill

Find a + b

Find 2 a

Find a + b

Trigonometry and VectorsVectors – Rectangular Components

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.

Trigonometry and VectorsVectors – Rectangular Components

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:

Vectors – Rectangular Components

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

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

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

cos = Fx / F

Fx = F cos i

sin = Fy / F

Fy = F sin j

What is the relationship between , sin , and cos ?

F Fx +

When are Fx and Fy Positive/Negative?

FFFx -Fy -

Fx +Fy -

Complete the following chart in your notebook:

III IV

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

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

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

Space Junk:

150 lb

iCosFx )60(*150

iiFx 752

jSinFy )60(*150

jFy 375

jiF 37575

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

Gravity

100 lb

jFy 100

jF 100

Trigonometry and Vectors2) Add up all forces in x direction

100 lb

150 lb

iFx 75

Trigonometry and Vectors3) Add up all forces in y direction

100 lb

150 lb

jjFy 100375

Trigonometry and Vectors4) Write resultant as single vector in

rectangular components

100 lb

150 lb

jiF 10037575

Classwork

Complete Worksheet

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

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

R = SFx + SFy

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

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

CLASSWORK / HOMEWORK

Complete problem #4 on the Vector Worksheet

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