TRIGONOMETRY

Consider a right angle triangle, ABC. The right angle is at C, and the lengths of the sides are labeled
a, b, and c . The Pythagorean theorem tells that the square of the hypotenuse (longest side)
is equal to the sum the the squares of the other two sides (called the legs ).


click here to practice: print-out page 1 and page 5 and solve. you need a calculator.


more practice . no picking !

ratios between the sides of a right triangle have given names: the trig functions.



Sine = Opposite side over Hypotenuse (S = O/H)
Cosine = Adjacent side over Hypotenuse (C= A/H)
Tangent = Opposite over Adjacent side (T=O/A)

REMEMBER: SOHCAHTOA


we can use trig to measure the height of a building. this will be your lab.
Find the height of the classroom. Use your height from eye level, the
distance to the wall and the angle between the horizontal and the
corner between the wall and the ceiling.

click here to practice . print the worksheet.

click here for more practice

more practice .

try that too

INTEGERS: REVIEW

To understand how to add or subtract vectors, review integers.
Horizontal and vertical number lines :

Try:
Along the horizontal:
5 - 7  =         plot the point
-3 + 5 =        plot the point
-1 - 2 =         plot the point
Along the vertical:
-4 + 9 =
9 - 12 =
-3 -1 =








VECTORS

A vector is a quantity that involves both amplitude (or magnitude) and direction
and obeys the communicative
law for addition. If A and B are vectors we have A + B = B + A
A scalar quantity is completely described by its size or magnitude.
In Physics, speed is a scalar (3m/s) and velocity is a vector (3m/s to South)
acceleration is a vector
distance is a distance (3 blocks) , displacement is a displacement (3 blocks to North)

see the figure below/
r is a vector. r could be a displacement if you were to walk from the
tail of the vector to the tip. the length of r could be proportional
(use a scale) to the distance you walked. r gives you also a direction.
The displacement vector tells us "how far" and "which way"
(example: r = 50m@20 degrees Northeast)

Using a protractor and a ruler, define the vector r:
r = _____@_________

scale: 1cm = 10m




r could be the velocity of you car ("how fast", "which way")
r could be the force acting on a plane (see figure above, the resultant of drag + thrust) ( "how strong", "which way")

In a x-y coordinate system, r_x is the horizontal displacement. r_x is called the x-component
r_y is the vertical displacement. r_y is called the y-component.
Can you see: r = r_x + r_y ? You are adding 2 displacements.

Can you find r_x and r_y using cosinus or sinus ? (and the angle)

r_{x = ______________}

r_{y = _______________}

We use vectors to represent displacement, velocity or force.
We use also the components of the vectors in mechanics a lot.

HOMEWORK I will collect one per table, my choice !

1) If you travel 350 miles in a direction 30 degrees North of East.
a) in a coordinate system x-y trace the vector displacement.
hint: The magnitude is 350 and the angle between EAst (positive x) and the vector is 30
b) How much is your displacement in the northward direction (that is the vertical component Sy )?
Use the sine function. You will need to trace the right angle triangle and to identify the opposite side...
c)How much is your displacement in the Eastward direction ? (find the horizontal component Sx).
Use the cosine function.

2)You are driving along at 20 m/s on a road that heads at an angle of 25 degrees N of W.
a) trace the vector in a coordinate system. Your vector velocity must be in the 2nd quadrant.
There is a 25 degrees angle between the vector and the west direction
b) At what rate are you going
(1) North
That is:Find the y component using sine
 (2) West ?
Find the x component using cosine

3) An airplane descending to the runway at 250 m/s is going at an angle of 22 degrees with the horizontal.
Find (a) its horizontal velocity  (horizontal component)
 (b) its rate of descent.(vertical component. It is a negative number, the plane is going down)
Your vector velocity is in the 4th quadrant.

You can do the opposite. Given the vector (displacement or other vector),
you can find the components.

4) In the following diagram, a force is acting on a box.
a)Draw each diagram on a graph paper using:
the first vector makes a 45 degrees angle with the horizontal
the second vector makes a 30 degrees with the horizontal
the third vector makes a 60 degrees with the horizontal
The 3 forces have a magnitude of 30N. use the scale 10N = 1 cm
b) find the x-component and the y-component
for each force. Fx = __________ and Fy = ______________
use cosine or sine.
c)for each vector check the pythagorian theorem:
F² = Fx² + Fy²

:


  worksheet or homework .  you need graph paper. try without reading the hints.

5) Make a sketch of each vector in standard position. 1.0 cm = 10m
The angle is always taken from the half axis ( 0, X ) or East direction. Open the angle counter clockwise
from the 1st quadrant to the others.

V= 20m @ 25 degrees 
T = 15m @ 105 degrees
M = 30m @ 285 degrees
N = 40m @ 190 degrees

6) Find the x-components and the y components of the vectors:
hint: to find the x-component use the cosine (X = cos(angle) x hyp)
                 to find the y-component use the sine (Y = sin(angle) x hyp)
This is only true if you are given the angle between the vector and the 0,x axis (East direction)

V= 20m @ 25 degrees
U = 25m @ 45 degrees
T = 15m @ 105 degrees
M = 30m @ 285 degrees


7) you are given the components of vectors.
Find the magnitude and the direction of the vectors.
Hint : to find the magnitude, use Pythagorean theorem.
to find the direction, find the angle between the half axis (0, X) (the EAst direction )and the vector.
a sketch would help you to locate the vector in one of the quadrant.

a) X = + 19.5 m    Y = - 49.6 m     (fourth quadrant) so  R = _____ @ _____     with R² = X² + Y²   angle = 360 degrees  - tan^-1 ( 49.6 / 19.5)
b) X= - 3240 ft         Y= - 1890ft  (third quadrant)  so R = _____ @ _____     with R² = X² + Y²  angle =180 degrees +  tan^-1 ( 3240 / 1890)  
c) X = - 158km         Y = 236 km (second quadrant)
d) X = 50 m              Y =  50m  

8) Use the scale 10 N = 1cm to trace the vectors:
hint: start from the origin. trace the x-component (horizontal component) then the y-component (vectical)
then connect the origin to the head of the vertical component. X + Y = R
  a) X = 30N   Y = 50N       (to the right and up)
b) X = - 40N    Y = - 10N  (to the left and down)
c) X = -80N Y = 50N

8 BIS)
a)An airplane pilot is traveling 500 miles per hour due South. He meets a wind current
that is traveling 75 miles per hour due west. To continue flying due south, what adjustment
in his navigation does he have to make ?
Hint: draw the vector . start from the origin. trace the x-component (75 mi due W) then the y-component (500 mi S)

b) A marathon runner runs 16 miles due north and then 5 miles due west.
What is his displacement ?  Displacement = _______ @ _______
hint: X = -5 and Y = 16  trace the vector in the second quadrant.



ADDING PARALLEL VECTORS



 Adding // vectors is easy. Remember the number line? 
A positive sign indicates the right and
a negative sign indicates the left. Add the integers together.
The addition of vectors is called the resultant.




9)You walk 3 blocks to the East (write A=+3)  and 8 blocks to the West (write B = -8).
What is you displacement ? (Find A + B).

10)a)You walk 3 miles to the North and 10 miles to the south what is your displacement ?

b)You walk 10 blocks East and 10 West ?

c)You walk 20 blocks East and 10 West ?

d) circle the vector quantities
acceleration, mass, temperature, velocity, energy, speed, density, area, distance,
time, displacement, volume, force


ADDING NON PARALLEL VECTORS USING THE TAIL-TO-HEAD METHOD

You walk 2 miles north and then 3 miles east , you have walked walked 5 miles.
But your displacement from the starting point is 3.6 miles east of north.
This quantity is called a vector sum of 2 vectors or the resultant.
Can you make a drawing?
answers

To add 2 vectors geometrically, use the tail-to-tip method.
Let's say you want to add A+B

Place the tail of B at the tip of A and connect them:



The sum of 2 or more vectors is called the resultant R. R = A + B in the previous example.
The x-component of the resultant is the sum of the sum of the x-components of each of the vectors you add.
The y-component of the resultant is the sum of the y-components of each of the vectors you add.
Rx = Ax + Bx
Ry = Ay + By
The components can be negative. (to the left = negative x-component, down = negative y-component).

click here you will find an applet that will help you understand vectors operations using components
practice: Get some graph papers and try to find A + B and A - B. Check your answers with the applet.

Using graph paper. do the worksheet  : adding vectors using components.
___________________________________________________________________________
worksheet:

A) Given vectors A, B, and C fine the x and y component of the resultant R 2 different ways.
 Follow the direction:
Move A so  its initial point (tail)  at the origin (0,) and // to its given position. To move it, use its components.
Next construct B with its tail on the head of vector A. Use its components. THen move The tail of C
on the head of B. Trace the resultant R. the tail of R is at the tail of A and its head at the head of C.
From the graph read the x and y components of R. Then check using the formula:
Rx = Ax + Bx + Cy and Ry = Ay + By + Cy



B)



ADDING NON PARALLEL VECTORS USING THE PARALLELOGRAM  METHOD


click here to understand the parallelogram method to add vectors.

MORE PROBLEMS TO PRACTICE (try without hints )

Try:  draw vector diagram for each problem. You can use the pythagorian theorem, the laws of sines or
cosines or geometry.

11. Consider a vector R. If Rx = 3 blocks and Ry = 10 blocks:
a) trace R in an x-y coordinate system . Use a graph paper. Hint: Start from the origin and move 3 blocks to the
right then move 10 blocks up. This point (3,10) shows the location of the head of the vector. Place the tail at
the origin and connect the origin (0,0) to (3,10) don't forget the head.
b)Find the magnitude (that is the length) of R using pythagorian theorem.
c) Find the direction of R using trig. You need to find the angle between R and the positive x-axis
 (that is the EAst direction)
d) finally write R = ____ @ _______


11 bis)You walk 7 miles south and then 3 miles west.
What is your displacement from your starting point ?
D = ____ @ _____
hint: find the magnitude of the displacement and its direction
If you work in the x-y system Dx = -3 and Dy = -7
start with the horizontal displacement then the vertical displacement and add the components to find
the vector D (or resultant)

12. A child is playing with a car on the floor of a train that is moving eastward.
While the train travels 12.0 m, the child pushes the car 2.6m northward on the floor
of  the train. What is the resulting displacement of the car ?
By now you should understand:
Dx = 12m and Dy = 2.6 m. You need to find the magnitude and the direction of the vector.
D = ____ @ ____

13.A 110N force and a 55N force act on a point P. The 110N force acts due North.
THe 55 N force acts due east . WHat is the magnitude and direction of the resultant force ?
F = ______ @ _____


14.

a boat can travel 4.0 m/s in still water. It is in a river that flows at 5.5 m/s southward.
If the boat heads eastward, directly across the river, what are the direction and
magnitude of its total velocity ?
Vx = 4 m/s
Vy = - 5.5 m/s
Start from the origin. move 4 to the right, 5.5 down. Connect (0,0) to (4,-5.5)
to find the resultant velocity. The head location is (4, -5.5)
Find the magnitude of the resultant using pythagorian and the direction
using tangent.





15. A plane is headed directly east at 340 mi/hr when the wind is from the south at 45 mi/hr.
What is its velocity (magnitude and direction) with respect to the ground ?

16. 2 soccers players kick the ball at exactly the same time. One player's foot exerts a force of 66N north.
The other's foot exerts a force of 88N east. What is the magnitude and direction of the resultant force on the ball ?

17.. A boat travels at 3.8 m/s and heads straight across a river 240m wide. The river flows at 1.6 m/s
What is the boat's resultant speed with respect to the bank ? How does it take the boat to cross the river ?
How far downstream is the boat when it reaches the other side ?

18. A bird is perched in the tree where it has its nest. The bird flies 500m due east and lands on the ground
in a field where it finds a worm. When the bird takes off, it is chased by a hawk, so the bird flies 300m
due North before landing in a tree. What direction must the bird fly to find its nest, and how far away
is the nest ? (please try without reading the hint)

hint: draw the  nest @ origin
the worm 500m @east is the x-component
the tree 300m@ north is the y-component
the resultant x-component + y-component is the vector connecting the nest  to the tree.  (sing tail to head)
the magnitude of the resultant is the distance you are looking for (use Pythagorean)
you need to find the direction of the resultant. (N of E or S of W or.....)


Hey ! you ! Can you help me finding my meal ?
show me the way using vectors , don't forget the
magnitude and the direction. the head and the tail !


18bis
click here you will find an applet that will help you understand vectors operations using components practice: Get some graph papers and try to find A + B and A - B.
Check your answers with the applet.

SUBTRACTING  NON PARALLEL VECTORS
To subtract one vector from another, for example, to get A - B, simply form the vector
- B, which is the scalar multiple of (-1) B, and add it to A.




try without hints:
19) In the 4 following problems (in the frame below the instructions , you have one problem per quadrant):
(use graph paper to trace the vectors)
i) trace U and V on a graph paper.
ii) find the components of U (Ux and Uy) and the components of V (Vx and Vy).
remember, the components can be negative. (left = negative, down = negative)
hints:start from the tail. move to the right or left  then up or down to get to the head.
Number of blocks you move along horizontal = x-component. Number of blocks you move along vertical = y component.
iii) Trace with a different color the vector R=U - V that is U + ( - V).
The size of (-V) and V is the same but they have opposite direction.
hints: first trace - V. Then move  (-V ) to the head of U (tail-to-head) . Trace the resulant by connecting the tail of U to the head of (-V).
iv) By looking at your diagram find the components of R.
hint: start from the tail of R and move to the head. the horizontal displacement = Rx, the vertical displacement = Ry.
v) Find the components of R using a mathematical method. instead of a geometrical method:
Rx = Ux + (- Vx)       add the components. you are adding integers. the components can be negative.
Ry = Uy + (- Vy)
Do you find the same thing? (hint: you should)



20) click here to understand the parallelogram method to add vectors or subtract.
if you subtract U - V add U + (-V) with (-V) having the same size but opposite direction as V.

LABS

material : forces table

part1   part2     part3    part4    part5    part6    part7


also: webcast: very advanced class on vectors. (MIT)