- #1
waqarrashid33 said:i tried a lot but answer goes wrong..
i didn't touched with calculus since long time...May be it is because of this...
For the integral of 1+cos(2t) gives 2T+sin(2T).
waqarrashid33 said:i tried a lot but answer goes wrong..
i didn't touched with calculus since long time...May be it is because of this...
For the integral of 1+cos(2t) gives 2T+sin(2T).
waqarrashid33 said:Thanks...
DonAntonio said:Interesting: you don't even need [itex]\,\,T\to\infty\,\,[/itex]. It is 1/2 for any [itex]\,\,T\neq 0\,[/itex].
DonAntonio
Mentallic said:Not quite, the final steps of the solution are to simplify [tex]\frac{1}{2}\left(1+\lim_{T\to a}\frac{\sin(2T)}{2T}\right)[/tex]
and that expression is only equal to 1/2 if [tex]\lim_{T\to a}\frac{\sin(2T)}{2T}=0[/tex] which only happens for [itex]a=\infty[/itex]
DonAntonio said:I don't know how you got that. I get
[tex]\int_{-T}^T \cos^2(t)dt=\left[\frac{t+\cos t\sin t}{2}\right]_{-T}^T[/tex]
DonAntonio said:[tex]\left(-T-\cos(-T)\sin(-T)\right)[/tex]
Mentallic said:How did you get that?
[tex]\cos^2t=\frac{1}{2}\left(1+\cos(2t)\right)[/tex]
Oh ok I see what you have, after integrating you converted sin(2t) to 2sin(t)cos(t)
This should be
[tex]\left(-T+\cos(-T)\sin(-T)\right)[/tex]
In this context, "half" refers to the quantity or amount that is equivalent to 1/2, or one half.
The concept of "half" has been used in various cultures and civilizations throughout history. The Ancient Egyptians, Babylonians, and Chinese all had their own systems of representing fractions, including 1/2. However, the modern notation for fractions, including 1/2, was developed by the ancient Greeks.
Different cultures have different ways of representing 1/2. For example, in the decimal system, 1/2 is represented as 0.5. In the Egyptian system, 1/2 was represented as a hieroglyphic symbol of a mouth. In the Mayan system, 1/2 was represented as a dot. The Chinese used a character for "half" to represent 1/2.
The mathematical definition of 1/2 is a fraction that represents one part of a whole that has been divided into two equal parts. It can also be represented as a decimal (0.5) or a percentage (50%). In terms of division, 1/2 is the result of dividing a number by 2.
Yes, 1/2 can be written in various forms, such as a decimal (0.5), a percentage (50%), a ratio (1:2), or a fraction in lowest terms (1/2). It can also be expressed as a mixed number (1/2 = 0.5 = 1/2 = 2/4 = 3/6 = ...).