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TRIGONOMETRIC RATIOS AND IDENTITIES

Angles and their measures:
* An angle is the union of two rays which have a common end point known as the vertex. The rays which form the angle are known as the arms of the angle.
* The measure of an angle is the least amount of rotation from the direction of one ray towards the direction of the other ray.
* An angle measured in anticlockwise sense is considered to be positive and an angle measured in clockwise sense is considered to be negative.


                       
Systems of measurement: An angle can be measured in the following three systems.

(i) Sexagesimal measure (British system)
(ii) Centesimal measure (French system)
(iii) Radian measure or Circular measure

 

(i) Sexagesimal measure: In this system a right angle is divided into 90 equal parts called degree. Each degree is sub divided into 60 equal parts called minutes and each minute is divided into 60 equal parts called seconds.
1 right angle = 90o
1o = 60'
1' = 60''

 

(ii) Centesimal measure: In this system a right angle is divided into 100 equal parts called grades. Each grade is subdivided into 100 equal parts known as minutes and each minute is sub divided into 100 equal parts known as seconds.
1 right angle = 100g (g = grades)
1g = 100'
1' = 100''
Note: One minute of centesimal system is different from one minute in sexagesimal system.
One second of sexagesimal system is different from one second of centesimal system.

 

(iii) Radian measure or Circular measure
* In this system an angle is measured in radians.
* Radian is the angle subtended by an arc of a circle at its centre whose length is equal to its radians.
l = r
* If l = r,
    = 1 radian = 1c
* 1 right angle =  
Relation between the three systems
If D, G and C are the measures of an angle in sexagesimal system, centesimal system and radian measure.
D= Gg = Cc
  right angles =   right angles =   right angles.


REGULAR POLYGON
* Angle subtended by each side of a regular polygon at its centre is 
* Each interior angle of a regular polygon is (n - 2) 
* Sum of interior angles of a regular polygon is (n - 2) πc
Note: Angle between any two sides of a regular polygon is known as its interior angle.

 

Trigonometric ratios of acute angles:


Consider a right angle ∆ABC
right angled at B. Let 
AB = h, BC = b, AC = l
Now we define six ratios by considering any two sides at a time. 


Where h = side opposite to
b = sides adjacent to
l = length of hypotenuse
Trigonometric Ratios of angles of any magnitude
The trigonometric ratios of angles of any magnitude can be defined using the concept of initial and terminal ray. A ray in the standard position is known as initial ray. After this ray is rotated by some angle it is known as terminal ray in its new position.
        Let the initial ray   be rotated by an angle in anticlockwise sense and be represented as  in its new position. This ray  , can lie in any of the four quadrants depending on the value of .

If 0o < < 90o,   is in first quadrant
If 90o < <180o,   is in second quadrant
If 180o < < 270o,   is in third quadrant
If 270o < < 360o,   is in fourth quadrant
       Let P(x, y) be any point on the terminal ray at a distance 'r' from the origin, then we define the six trigonometric ratios as

                           


Sign of trigonometric ratios in various quadrants

     
Trigonometric ratios of standard angles:

 

     
                                           n.d.  not defined (± ∞)
Increasing and decreasing nature of trigonometric functions in various quadrants


↑ increases  ↓decreases

 

Domain and Range of trigonometric functions:

Posted Date : 19-02-2021

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