Consider a unit circle with center at the origin O and Let P be any point on the circle with P(a,b). And let call the angle x
We use the coordinates of P to define the cosine of the angle and the sine of the angle. Specifically, the x-coordinate of B is the cosine of the angle, and the y-coordinate of B is the sine of the angle.
Also it is clear

a^{2}+b^{2}=1

cos^{2}x +sin^{2}x =1

- Sine and cosine are periodic functions of period $360^{\circ}$, that is, of period $2\pi $.
That's because sines and cosines are defined in terms of angles, and you can add
multiples of $360^{\circ}$, or $2\pi $, and it doesn't change the angle. Thus, for any angle x

sin(x+2π)= sin (x) and cos(x+2π)= cos(x)

or we can say that

sin (2nπ + x) = sin x, $n\in Z$ , cos (2nπ + x) = cos x, $n\in Z$

Where Z is the set of all integers - sin x = 0 implies x = nπ, where n is any integer

cos x = 0 implies x = (2n + 1)(π/2) - The other trignometric function are defined as

cosec(x)= 1/sin (x) where x ≠nπ, where n is any integer

sec(x)=1/cos (x) where x ≠(2n + 1)(π/2) where n is any integer

tan(x)=sin(x)/cos(x) where x ≠(2n + 1)(π/2) where n is any integer

cot(x)=cos(x)/sin(x) where x ≠nπ, where n is any integer - For all real x

sin^{2}(x)+cos^{2}(x)=1

1+ tan^{2}(x)=sec^{2}(x)

1+ cot^{2}(x)=cosec^{2}(x) - What is is Odd function and Even Function

We have come across these adjectives 'odd' and 'even' when applied to functions, but it's important to know them. A function f is said to be an odd function

if for any number x,*f(-x) = -f(x)*.

A function f is said to be an even function if for any number x,*f(-x) = f(x)*.

Many functions are neither odd nor even functions, but some of the most important functions are one or the other.

Example:

Any polynomial with only odd degree terms is an odd function, for example, f(x) = 2x^{7}+ 9x^{5}- x. (Note that all the powers of x are odd numbers.)

Similarly, any polynomial with only even degree terms is an even function. For example, f(x) = 6x^{8}- 6x^{2}- 5.

Based on above defination we can call Sine is an odd function, and cosine is even

sin (-x) = -sin x, and

cos (-x) = cos x.

These facts follow from the symmetry of the unit circle across the x-axis. The angle -x is the same angle as x except it's on the other side of the x-axis. Flipping a point (x,y) to the other side of the x-axis makes it into (x,-y), so the y-coordinate is negated, that is, the sine is negated, but the x-coordinate remains the same, that is, the cosine is unchanged. - Now since in unit circle

-1 ≤ a ≤ 1

-1 ≤ b ≤ 1

It follows that for all x

-1 ≤ sin(x) ≤ 1

-1 ≤ cos(x) ≤ 1 Also We know from previous classes,

a,b are both positive in Ist quadrant i.e 0< x < π/2 It implies that sin is positive and cos is postive

a is negative and b is positive in IInd quadrant i.e π/2 < x< πIt implies that sin is negative and cos is postive

a and b both are negative in III quadrant ie. π < x < 3π/2 It implies that sin is negative and cos is negative

a is positive and b is negative in IV quadrant i,.e 3π/2 < x < 2π It implies that sin is positive and cos is negative

Similarly sign can be obtained for other functions

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