Class 11 Maths- Trignometry functions

Trignometric functions:

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

a²+b²=1

cos²x +sin²x =1

Properties of these functions

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
2. sin x = 0 implies x = nπ, where n is any integer
   cos x = 0 implies x = (2n + 1)(π/2)
3. 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
4. For all real x
   sin²(x)+cos²(x)=1
   1+ tan²(x)=sec²(x)
   1+ cot²(x)=cosec²(x)
5. 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) = 2x⁷ + 9x⁵ - 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) = 6x⁸ - 6x² - 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.
6. 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