Assalamualaikum..This animal(menatang ni) aku rasa hampa smua besa dok baca kot, kan kan?. Jadi aku saja post cuma tambah poster tu ja. hahaha.
Kalau lelaki handsome pendiam
Perempuan akan cakap: ciri-ciri lelaki idaman..
kalau lelaki tak handsome pendiam
Perempuan akan cakap: patutla, dah tak hensem..
kalau lelaki handsome berbuat jahat
Perempuan akan kata: nobody’s perfect
kalau lelaki tak handsome berbuat jahat
perempuan akan cakap: memang…. muka pun macam pecah rumah!…..
kalau lelaki handsome menolong perempuan yg diganggu
perempuan akan cakap: wah.. machonya.. macam hero filem!
kalau lelaki tak handsome menolong perempuan yang diganggu
Perempuan akan kata: nak ambik kesempatan rr tu..
kalau lelaki handsome dapat perempuan cantik
perempuan akan kata: sepadan sangat…
kalau lelaki tak handsome dapat perempuan cantik
perempuan akan kata: mesti kena bomoh perempuan tuh!
kalau lelaki handsome ditinggal kekasih
perempuan akan kata: jangan sedih, kan saya ada..
kalau lelaki tak handsome ditinggal kekasih
perempuan akan kata:…(terdiam, tapi telunjuknyameliuk-liuk dari atas ke bawah, patutlah, tengok saja luarannya)…
kalau lelaki handsome penyayang binatang
perempuan akan cakap: perasaannya halus…penuh kasih sayang
kalau lelaki tak handsome penyayang binatang
perempuan akan cakap: sesama keluarga memang harus menyayangi…
kalau lelaki handsome bawa BMW
perempuan akan cakap: matching… hebat luar dalam
kalau lelaki tak handsome bawa BMW
perempuan akan cakap: bang, bosnya mana?…
kalau lelaki handsome tak mau bergambar
perempuan akan cakap: pasti takut kalau2 gambarnya tersebar
kalau lelaki tak handsome tak mau bergambar
perempuan akan kata: mesti sbb takut nak tgk muka sndri..
kalau lelaki handsome menuang air ke gelas perempuan
perempuan akan cakap: ini barulah lelaki gentlemen
kalau lelaki tak handsome menuang air ke gelas perempuan
perempuan akan cakap: naluri pembantu, memang begitu….
kalau lelaki handsome bersedih hati
perempuan akan cakap: let me be your shoulder to cry on
kalau lelaki tak handsome bersedih hati
perempuan akan kata: kuat nangis!! lelaki ke tak ni?
Tuesday, August 5, 2008
Sunday, July 27, 2008
Complex Plane
The Complex Plane
Complex numbers are points in the plane
In the same way that we think of real numbers as being points on a line, it is natural to identify a complex number z=a+ib with the point (a,b) in the cartesian plane. Expressions such as ``the complex number z'', and ``the point z'' are now interchangeable.
We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of the complex numbers. The reals are just the x-axis in the complex plane.
The modulus of the complex number z= a + ib now can be interpreted as the distance from z to the origin in the complex plane.
Since the hypotenuse of a right triangle is longer than the other sides, we have
for every complex number z.
We can also think of the point z= a+ ib as the vector (a,b). From this point of view, the addition of complex numbers is equivalent to vector addition in two dimensions and we can visualize it as laying arrows tail to end. (Picture)
We see in this way that the distance between two points z and w in the complex plane is z-w.
Exercise: Prove this last statement algebraically. (Proof.)
Exercise: Prove the ``Parallellogram law''
(Proof.)
The ``Triangle'' inequality
is easily seen to hold. (Proof.)
Exercise: Prove the Triangle inequality for n complex numbers
(Proof.)
Complex numbers are points in the plane
In the same way that we think of real numbers as being points on a line, it is natural to identify a complex number z=a+ib with the point (a,b) in the cartesian plane. Expressions such as ``the complex number z'', and ``the point z'' are now interchangeable.
We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of the complex numbers. The reals are just the x-axis in the complex plane.
The modulus of the complex number z= a + ib now can be interpreted as the distance from z to the origin in the complex plane.
Since the hypotenuse of a right triangle is longer than the other sides, we have
for every complex number z.
We can also think of the point z= a+ ib as the vector (a,b). From this point of view, the addition of complex numbers is equivalent to vector addition in two dimensions and we can visualize it as laying arrows tail to end. (Picture)
We see in this way that the distance between two points z and w in the complex plane is z-w.
Exercise: Prove this last statement algebraically. (Proof.)
Exercise: Prove the ``Parallellogram law''
(Proof.)
The ``Triangle'' inequality
is easily seen to hold. (Proof.)
Exercise: Prove the Triangle inequality for n complex numbers
(Proof.)
Saturday, July 26, 2008
Trigo.....!
TRIGONOMETRY - MEASURE OF AN ANGLE
Any real number may be interpreted as the radian measure of an angle as follows: If , think of wrapping a length of string around the standard unit circle C in the plane, with initial point P(1,0), and proceeding counterclockwise around the circle; do the same if , but wrap the string clockwise around the circle. This process is described in Figure 1 below.
Figure 1
If Q(x,y) is the point on the circle where the string ends, we may think of as being an angle by associating to it the central angle with vertex O(0,0) and sides passing through the points P and Q. If instead of wrapping a length s of string around the unit circle, we decide to wrap it around a circle of radius R, the angle (in radians) generated in the process will satisfy the following relation:
Observe that the length s of string gives the measure of the angle only when R=1.
As a matter of common practice and convenience, it is useful to measure angles in degrees, which are defined by partitioning one whole revolution into 360 equal parts, each of which is then called one degree. In this way, one whole revolution around the unit circle measures radians and also 360 degrees (or ), that is:
Each degree may be further subdivided into 60 parts, called minutes, and in turn each minute may be subdivided into another 60 parts, called seconds:
EXAMPLE 1 Express the angle in Degree-Minute-Second (DMS) notation.
Solution: We use Equation 3 to convert a fraction of a degree into minutes and a fraction of a minute into seconds:
Therefore, .
EXAMPLE 2 Express the angle in radians.
Solution: From Equation 2 we see that
EXAMPLE 3 Find the length of an arc on a circle of radius 75 inches that spans a central angle of measure .
Solution: We use Equation 1, , with R=75 inches and , to obtain
Here are some more exercises in the use of the rules given in Equations 1,2, and 3.
EXERCISE 1 Express the angle radians in (a) decimal form and (b) DMS form.
Solution
EXERCISE 2 Express the angle in radians.
Solution
EXERCISE 3 Assume that City A and City B are located on the same meridian in the Northern hemisphere and that the earth is a sphere of radius 4000 mi. The latitudes of City A and City B are and , respectively.
(a)
Express the latitudes of City A and City B in decimal form.
(b)
Express the latitudes of City A and City B in radian form.
(c)
Find the distance between the two cities.
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