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trignometry




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




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