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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
IOT
POLY ENGINEERING3-8
Trigonometry and Vectors
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
IOT
POLY ENGINEERING3-9
Trigonometry and Vectors
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
IOT
POLY ENGINEERING3-9
Trigonometry and Vectors
1. 5 miles northeast
2. 6 yards
3. 1000 lbs force
Scalar or Vector?
VectorMagnitude and Direction
ScalarMagnitude only
ScalarMagnitude only
4. 400 mph due north
5. $100
6. 10 lbs weight
VectorMagnitude and Direction
ScalarMagnitude only
VectorMagnitude and Direction
IOT
POLY ENGINEERING3-9
Trigonometry and Vectors
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
IOT
POLY ENGINEERING3-9
Trigonometry and Vectors
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
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
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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POLY ENGINEERING3-10
Trigonometry and Vectors
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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POLY ENGINEERING3-10
Find 2 a
March 14, 2011Drill
y
x
ay
x
a
b
Find a + b
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POLY ENGINEERING3-10
Find 2 a
Drill
y
x
a
2a
IOT
POLY ENGINEERING3-10
Drill
y
x
a
b
Find a + b
aa+b
IOT
POLY ENGINEERING3-10
Drill
y
x
a
b
Find a + b
a+b
b
IOT
POLY ENGINEERING3-10
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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POLY ENGINEERING3-10
Trigonometry and VectorsVectors – Rectangular Components
y
x
F
Fx
Fy
This means:
vector F = vector Fx + vector Fy
Remember the addition of vectors:
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
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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POLY ENGINEERING3-10
Trigonometry and Vectors
Vectors – Rectangular Components
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
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
Vectors – Rectangular Components
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
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
Vectors – Rectangular Components
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 ?
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
Vectors – Rectangular Components
y
x
F Fx +
Fy +
When are Fx and Fy Positive/Negative?
FFx -
Fy +
FFFx -Fy -
Fx +Fy -
IOT
POLY ENGINEERING3-10
Vectors – Rectangular Components
Complete the following chart in your notebook:
III
III IV
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POLY ENGINEERING3-10
Trigonometry and Vectors
Vectors – Rectangular Components
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
IOT
POLY ENGINEERING3-10
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
IOT
POLY ENGINEERING3-10
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
IOT
POLY ENGINEERING3-10
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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POLY ENGINEERING3-10
Trigonometry and Vectors2) Add up all forces in x direction
100 lb
150 lb
60o
iFx 75
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors3) Add up all forces in y direction
100 lb
150 lb
60o
jjFy 100375
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors4) Write resultant as single vector in
rectangular components
100 lb
150 lb
60o
jiF 10037575
IOT
POLY ENGINEERING3-10
Classwork
Complete Worksheet
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POLY ENGINEERING3-10
Trigonometry and VectorsVectors – Resultant Forces
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
IOT
POLY ENGINEERING3-10
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
Trigonometry and VectorsVectors – Resultant Forces
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
IOT
POLY ENGINEERING3-10
Trigonometry and VectorsVectors – Resultant Forces
R = SFx + SFy
R = (75 lbs)i + (75 - 100 lbs)j
R = (75 lbs)i + (29.9 lbs)j
3
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
Trigonometry and Vectors
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
IOT
POLY ENGINEERING3-10
Trigonometry and Vectors
CLASSWORK / HOMEWORK
Complete problem #4 on the Vector Worksheet
IOT
POLY ENGINEERING3-10