Download direct proportional

www.aoua.com/vb
UNIT 13
direct proportional
relationship between a & p, and this relation is denoted by:
a ∞p
This is equivalent to a = k p or a/p = k
Where k is the constant of proportionality.
Cubic proportional relationship
In general: If you scale up all the dimensions of any three
dimensional
solid by the same factor, you scale the volume by the cube of
that factor.
Example: If you have a recipe for a birthday cake which fits a
square tin with dimensions 10×10×5, and you want to rescale the
ingredients to fit a square tin with dimensions 5×5×2.5, then you
need to divide all the ingredients by (2)3, that is 8.
The volume is directly proportional to the cube of one of its
dimensions, x say. So: V ∞ x3 or V = kx3
The graphs of cubic proportional relationship are not straight
lines.
Square proportional relationship
In general: If you scale up all the dimensions of any shape by
the same factor, you scale the area by the square of that
factor.
Example: If you need to paint a square wall with side 6m, and the
tank of paint covers a square with side 3m, then you need 4 tanks.
The area is directly proportional to the square of one
dimension of the shape, x say. So: Area ∞ x2 or Area = kx2
The graphs of square proportional relationship could be
represented
parabolas.
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 1
Sherif. GOOOOOOD LUCK TO ALL
www.aoua.com/vb
Inverse proportional relationship:
When y decreases as x increases, then y is inversely
proportional to x.
Symbolically, y ∞ 1/x or y = k/x
Period of the sum of two sines functions
The period of the sum of the two sines is 2π except
where the two coefficients have common factor , in
which case the period (2π)is divided by that common
factor .
Examples
What is the period of the following functions :
(a) y = sin 5x + sin 15x
Answer
Period = 2π/5
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 2
Sherif. GOOOOOOD LUCK TO ALL
www.aoua.com/vb
UNIT 14
For two similar shapes, we have two relationships:
• The scale factor relationship between the two shapes; that is,
The length of a side of one shape =
scale factor × the length of the corresponding side of the other
shape.
For example, for the above triangles,
a' = scale factor × a & b' = scale factor × b & c' = scale factor × c
or this could be written as: =
a\ = b\
a
b
=
c\
c
=
scale factor
The ratio of the sides within one shape is the same as the ratio
of the corresponding sides within the other shape.
Notes:
1- Similarity between shapes occurs also by angles .
In another way:
 When two angles in one triangle are equal to the two
angles in the other triangle then the triangles are similar.
2- If two triangles have one identical angle, and the sides
containing that angle are in the same ratio in both
triangles then, the triangles will be similar.
Similarity between Circles
You must know that any two circles are similar.
 Area of Circle = π r ²
 Circumference of circle (c) = 2 π r
Also Diameter (d) = 2 r
The same .
Circumference of circle = π D
Consider a circle whose diameter of length of 1 unit:
Then it's circumference is:
C =2πr =πD
C =πx1
C= π
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 3
Sherif. GOOOOOOD LUCK TO ALL
www.aoua.com/vb
Trigonometry
How to find angles:
In trigonometry … π = 180°
Radian = θ x ( π /180
Calculate angles by using triangles
A
opposite
Hypotenuse
B
Sin θ =
Cos θ =
Opposite
Hypotenuse
Adjacent
Hypotenuse
Opposite
Adjacent
Other rules:
tan θ =
Cosec θ = 1
Sinθ
Adjacent
C
Sin ratio
Cos ratio
Tangent ratio
Sec θ =
1
Cosθ
Cotan θ = 1
tanθ
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 4
Sherif. GOOOOOOD LUCK TO ALL
www.aoua.com/vb
Note:
You have to know that the summation of all Angles of any
triangle is equal to 180º.
not ( 90°) ‫و لكن اذا كان المثلث غير قائم‬
Use either sin rule or cos rule
Use the sine rule:
Sin A
a
= Sin B = Sin C
b
c
Sine rule occurs only if we have minimum 2 angles and one
side.
‫الزم يكون عندى زاويتين و طول ضلع واحد‬
Use the cosine rule:
a² = b² + c² – 2bc cos A
Sides
b² = a² + c² – 2ac cos B
c² = a² + b² – 2ab cos C
Angles
Cos A = b² + c² – a²
2bc
Cos B = a² + c² – b²
2ac
Cos C = a² + b² – c²
2ab
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 5
Sherif. GOOOOOOD LUCK TO ALL
www.aoua.com/vb
UNIT 15
Amplitude (A), plotted on the vertical axis, is the maximum
height of the sine curve
above or below the horizontal axis.
Period (T), plotted on the horizontal axis, is the time required
for one complete cycle.
Frequency (f) is the number of complete cycles per seconds
(often measured by hertz).
Period and frequency are inversely related, so f = 1/T and T =
1/f.
The angular frequency (ω) is the frequency multiplied by 2 π,
that is ω = 2 π f.
Three alternative ways for writing the formula for a sine
curve with amplitude A
1- If you know the period, then the variation of y with time t is
described by the formula Y=A sin (2 п / T *t)
2- If you know the frequency (in hertz), then the formula can be
written as:
Y=A sin (2 п f t)
3- If you know the angular frequency, then the formula can be
written as
y = Asin(ω t)
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 6
Sherif. GOOOOOOD LUCK TO ALL
www.aoua.com/vb
Sunrise, sunset formula, …
Y= M+ A sin (2 Л/T*t+φ)
Where: y is the time of sunset.
t is time of year (in weeks)
M is the mean or average value of the sunset time.
A is the amplitude of the sinusoidal model
T is the period
φ is the phase shift (positive or negative and should be in radians)
(Remember to convert minutes to hours with decimal
places).
Repeating patterns
When adding two sine curves, the result is another periodic curve
with frequency equal to the difference of the two curves
frequencies (in hertz).
To change from degree to radian:
R= D* Л/180
To convert a degree into a distance in kilometers:
1 degree= 111.14 km
The formula of knowing the distance between 2 points on earth:
Cos C= sin (LT1)sin(LT2)+ cos(LT1)cos(LT2) cos (LG1-LG2)
LT1= Latitude of the first point
LG1=Longtitude of the first point
LT2= Latitude of the second point
LG2=Longtitude of the second point
Notes:
* Read also Fourier's formula in Miss ferzana summary of unit 15.
* Go to calculator book and read carefully Chapter 14 p. 218
*Enter the programmes mentioned in this section.
Go to page 300 Ex. 14.5 and enter the great circle prog. GRTCIR
Prepared by Mujtahidah from the summaries of Miss Ferzana and Mr. 7
Sherif. GOOOOOOD LUCK TO ALL