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Identities without variables
The curious identity
is a special case of an identity that contains one variable:
The same cosine identity in radians is
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function
evaluated at (in the very last case above) 21; only half of the zeroes
are present above. The two identities preceding this last one arise in
the same fashion with 21 replaced by 10 and…
Beside the Cartesian , there are other systems also for representing points and curve analytically . Polar system which is one of them .
---------In this system we started with a fixed line OX called the Initial Line and a fixed point on it , called the Pole .
---------If P be any given point, the distance OP=r is called the Radius Vector and Angle XOP=theta the Vectorial Angle . The two together are referred to as the Polar Co-ordinates of P .
Unrestricted Variation of Polar Co-ordinates :-
If we concerned with assigning polar co-ordinates to only individual points in the plane , then it would clearly be enough to consider the radius vector to have positive values only and the vectorial angle theta to lie between 0 and 2pi .
Transformation of Co-ordinates :-
Take the initial line OX of the polar system as the positive direction of X-axis and the Pole O as origin for the Cartesian system .The positive direction of Y-axis is to be such …