IIT-JEE MATHEMATICS: notes for trigonometry

trignometry

1. Introduction

Trigonometry (from Greek trigõnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3^rd century BC from applications of geometry to astronomical studies.
Trigonometry is most simply associated with planar right angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles

What is angle

An angle which has its vertex at the origin and one side lying on the positive x-axis. It can have a measure which positive or negative and can be greater than 360°

If the direction of rotation is anticlockwise, angle is positive . If the direction of rotation is clockwise,angle is negative
Once you have made a full circle (360°) keep going and you will see that the angle is greater than 360° .In fact you can go around as many times as you like. The same thing happens when you go clockwise. The negative angle just keeps on increasing
It can be measured in degrees or radian

Degree and Radian

They both are unit of measurement of angles
Radian: A unit of measure for angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.
Degree: If a rotation from the initial side to terminal side is (1/360) of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is called a minute, written as 1", and one sixtieth of a minute is called a second, written as 1'.
Thus, 1° = 60', 1' = 60"
Relation between Degree and Radian
2π radian = 360 ° π radian= 180 ° 1 radian= (180/π) °

Degree	30°	45°	60°	90°	120°	180°	360°
Radian	π/6	π/4	π/3	π/2	2π/3	π	2π

Trigonmetric Ratio's

In a right angle triangle ABC where B=90° ,we can define six ratio's for the two sides i.e Hypotenuse/Base, Base/Perpendicular,Perpendicular/Base,Base/hypotenuse,Hypotenuse/Perpendicular,Perpedicular/Hyptenuse Trignometric ratio's are defined as
sin θ= Perpendicular/Hypotenuse
cosec θ= Hypotenuse/Perpendicular
cos θ= Base/Hypotenuse
sec θ= Hypotenuse/Base
tan θ= Perpendicular/Base
cot θ= Base/Perpendicular
Notice that each ratio in the right-hand column is the inverse, or the reciprocal, of the ratio in the left-hand column.
The reciprocal of sin θ is csc θ ; and vice-versa.
The reciprocal of cos θ is sec θ.
And the reciprocal of tan θ is cot θ
These are valid for acute angles.
We are now going to define them for any angles and they are called now the Trigometric 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

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
sin²(x)+cos²(x)=1
1+ tan²(x)=sec²(x)
1+ cot²(x)=cosec²(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) = 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.
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
Domain and Range of trigonimetric functions
1. y=f(x)= Sin(x)
  Domain : It is defined for all real values of x
  Range : -1 ≤ y ≤ 1
  Period:2π
  It is a odd function
2. y=f(x)= cos(x)
  Domain : It is defined for all real values of x
  Range : -1 ≤ y ≤ 1
  Period:2π
  It is even function
3. ) y=f(x)=tan(x)
  Domain : It is defined for all real values of x except x ≠(2n + 1)(π/2) where n is any
  Range : All the real numbers
  Period:π
  It is a odd function
4. y=f(x)=cot(x)
  Domain : It is defined for all real values of x except x ≠nπ, where n is any integer
  Range : All the real numbers
  Period:π
  It is a odd function
5. y=f(x)=sec(x)
  Domain : It is defined for all real values of x except x ≠(2n + 1)(π/2) where n is any integer
  Range : (-∞,-1] ∪ [1,∞)
  Period:2π
  It is even function
6. y=f(x)=cosec(x)
  Domain :It is defined for all real values of x except x ≠nπ, where n is any integer
  Range : (-∞,-1] ∪ [1,∞)
  Period:2π
  It is odd function