Properties of trigonometric functions

Properties of trigonometric functions:

It is often useful to remember and use the properties of trigonometric functions while applying trigonometry in real life.

The main properties among the properties of trigonometric functions are given below:


a> Quadrant rule of signs:

In first quadrant both abscissa and ordinate are positive or , y > 0 , x>0 , r>0  , so sine , cosine , tangent and all other trigonometric functions in this quadrant are positive.

But in second quadrant y > 0 , x<0 , r>0 thus in second quadrant:

 \sin \theta = \frac{y}{r} > 0 , \cos \theta = \frac{x}{r} < 0 , \tan = \theta = \frac{y}{x} = 0

and we can also analyze third and fourth quadrant similarly and we can conclude:


Sine is positive in first and second quadrant.

Cosine is positive in first and fourth quadrant.

Tangent is positive in first and third quadrant.


Which we can summarize in the picture below:


properties of trigonometric functions

properties of trigonometric functions


One popular method to remember this property  is called ( CAST ) with the understanding that only “C” or Cos and it’s reciprocal (Sec) are positive in fourth quadrant , “A” All ratios are positive in first quadrant , only “S” Sine and Cosecant ( reciprocal on sine) are positive in second quadrant and only “T” Tangent and cotangent(reciprocal of tangent) are positive in   third quadrant , as shown in the picture below:

properties of trigonometric functions

properties of trigonometric functions



b> Periodicity:

The trigonometric functions are periodic , that is , there is regular repetition of the values of the functions over a certain interval of angle.

For example: , as , \theta increases from o^\circ to 360^\circ ( 2\pi ) , the terminal arm completes one circle about the origin and occupies the same original position. ( Please see trigonometric functions if you are unable to understand the concept of terminal arm and circle ) , this is how trigonometric functions are periodic.

The sine and cosine functions are periodic with 2 \pi or ,

\sin (\theta + x.2\pi ) = \sin \theta ( X is any integer)

The Tangent and Cotangent function are periodic with \pi or,

\tan (\theta + x.\pi ) = \tan \theta ( X is any integer)


c> Limits of values of trigonometric functions:


The values of trigonometric functions ( sine and cosine) has a certain limit which is:

 - 1 \le \sin \theta \le 1


 - 1 \le \cos \theta \le 1

The Tangent function can limit from negative infinity to positive infinity , or it don’t have any limit.


d> Even and Odd function:


A function is said to be even if f ( -x ) = f(x) , and odd if f ( -x ) = - f(x) .

The cosine function is even but the sine function is odd.


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