Notes On Trigonometric Functions (C - Level) - CBSE Class 11 Maths
 
In a right angled triangle, the ratio of any two sides in a particular order gives rise to six different trigonometric ratios for the angle, q.
 

 
Consider a unit circle such that its centre is at the origin.
 
The circle intersects the X-axis at points A (1, 0) and C (-1, 0), and the Y-axis at points B (0, 1) and D (0,).


Let P (a, b) be any point P in the first quadrant such that ∠AOP= radians.


In a circle of radius r, if an arc of length l subtends an angle  radians at the centre, then .
 
Using the formula, we get the length of arc AP as.   
 

 
Draw a perpendicular from point P such that it intersects the X-axis at M.


Point M is at a distance of ‘a’ units from the origin on the X-axis and length MP is b units.


Also, OP being the radius of the unit circle is equal to 1.

∆OMP is a right-angled triangle.
Thus, in ∆OMP,

 …(i)


 Substituting the values of a and b in equation (i), .
This shows a relation between the sine and the cosine functions for any real q.
Quadrantal Angles: Angles that are integral multiples of  are called quadrantal angles.
When ,  




When ,  




When   




When ,  




Note: If we increase or decrease the value of q by an integral multiple of 2, then the values of sin q and cos q do not change.


, where n is any integer.
 
Or sin  0, if  =
 
, where n is any integer.
 
Or   0, if  =
 
Thus, sin q = 0 implies q =n 
 
Cos q = 0 implies q = (2n+1), where n is any integer.



Trigonometric functions in terms of sine and cosine functions
 
Tan  =   , where n is any integer


         [Cos q = 0 ⇒q = (2n+1), where n is any integer]


 Cot  = , ,  where n is any integer


         [Sin q = 0 ⇒q =n]


Sec  = , ,  where n is any integer


Cosec  = , ,  where n is any integer
 
Identities:
 


1 + =


1 + Cot2 = Cosec2
 
The table showing the values of trigonometric functions for some standard angles:
 

 
 
Signs of Trigonometric Functions:
 
Consider a unit circle such that its centre is at the origin. The circle intersects the X-axis at points A and C, and the Y-axis at points B and D.

 
The coordinates of Q are (a, -b).


For any point P(a, b),






Sin (-  


[


[


Similarly, cos (-  

[




[


Now, tan (-  


=


 Tan (-


Similarly, cot (-


 Sec (-


 Cosec (-


For any arbitrary point P(a, b),




.
 
Signs of trigonometric functions in different quadrants
 
In the first and the second quadrants, y coordinate is positive, which implies that the sine function is positive in the first two quadrants. In the third and the fourth quadrants, the y coordinate is negative, and hence, the sine function is negative.
Similarly, x coordinate of point P is negative in the second and the third quadrants, and therefore, the cos function is negative in these quadrants. Similarly, x coordinate is positive in the first and the fourth quadrants, and therefore, the cos function is positive in these quadrants.
 
Signs of the other four functions in different quadrants can be found in the same method.
 

 
The sign of a trigonometric function depends upon the quadrant in which a point lies.
  • In Quadrant I, all the trigonometric functions are positive.
  • In Quadrant II, the Sine and Cosecant functions are positive.
  • In Quadrant III, the Tan and Cot functions are positive.
  • In Quadrant IV, the Cos and Secant functions are positive.
 

 
A trick to remember this would be to remember the phrase “All Silver Tea Cups” or “A Smart Trig Class” or “Add Sugar To Coffee” starting in the first quadrant.
 

Summary

 
In a right angled triangle, the ratio of any two sides in a particular order gives rise to six different trigonometric ratios for the angle, q.
 

 
Consider a unit circle such that its centre is at the origin.
 
The circle intersects the X-axis at points A (1, 0) and C (-1, 0), and the Y-axis at points B (0, 1) and D (0,).


Let P (a, b) be any point P in the first quadrant such that ∠AOP= radians.


In a circle of radius r, if an arc of length l subtends an angle  radians at the centre, then .
 
Using the formula, we get the length of arc AP as.   
 

 
Draw a perpendicular from point P such that it intersects the X-axis at M.


Point M is at a distance of ‘a’ units from the origin on the X-axis and length MP is b units.


Also, OP being the radius of the unit circle is equal to 1.

∆OMP is a right-angled triangle.
Thus, in ∆OMP,

 …(i)


 Substituting the values of a and b in equation (i), .
This shows a relation between the sine and the cosine functions for any real q.
Quadrantal Angles: Angles that are integral multiples of  are called quadrantal angles.
When ,  




When ,  




When   




When ,  




Note: If we increase or decrease the value of q by an integral multiple of 2, then the values of sin q and cos q do not change.


, where n is any integer.
 
Or sin  0, if  =
 
, where n is any integer.
 
Or   0, if  =
 
Thus, sin q = 0 implies q =n 
 
Cos q = 0 implies q = (2n+1), where n is any integer.



Trigonometric functions in terms of sine and cosine functions
 
Tan  =   , where n is any integer


         [Cos q = 0 ⇒q = (2n+1), where n is any integer]


 Cot  = , ,  where n is any integer


         [Sin q = 0 ⇒q =n]


Sec  = , ,  where n is any integer


Cosec  = , ,  where n is any integer
 
Identities:
 


1 + =


1 + Cot2 = Cosec2
 
The table showing the values of trigonometric functions for some standard angles:
 

 
 
Signs of Trigonometric Functions:
 
Consider a unit circle such that its centre is at the origin. The circle intersects the X-axis at points A and C, and the Y-axis at points B and D.

 
The coordinates of Q are (a, -b).


For any point P(a, b),






Sin (-  


[


[


Similarly, cos (-  

[




[


Now, tan (-  


=


 Tan (-


Similarly, cot (-


 Sec (-


 Cosec (-


For any arbitrary point P(a, b),




.
 
Signs of trigonometric functions in different quadrants
 
In the first and the second quadrants, y coordinate is positive, which implies that the sine function is positive in the first two quadrants. In the third and the fourth quadrants, the y coordinate is negative, and hence, the sine function is negative.
Similarly, x coordinate of point P is negative in the second and the third quadrants, and therefore, the cos function is negative in these quadrants. Similarly, x coordinate is positive in the first and the fourth quadrants, and therefore, the cos function is positive in these quadrants.
 
Signs of the other four functions in different quadrants can be found in the same method.
 

 
The sign of a trigonometric function depends upon the quadrant in which a point lies.
  • In Quadrant I, all the trigonometric functions are positive.
  • In Quadrant II, the Sine and Cosecant functions are positive.
  • In Quadrant III, the Tan and Cot functions are positive.
  • In Quadrant IV, the Cos and Secant functions are positive.
 

 
A trick to remember this would be to remember the phrase “All Silver Tea Cups” or “A Smart Trig Class” or “Add Sugar To Coffee” starting in the first quadrant.
 

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