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Domain and Range of Trigonometric Functions

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Domain and Range of Trigonometric Functions - Lesson Summary

 
An interval in which the end points are included is called a closed interval. If one of the end points is infinity or minus infinity, even then the interval is a closed interval.
 
An interval is said to be a half-closed or a half-open interval if only one of the end points is included. There are two types of half-open or half-closed intervals.
 
An interval in which both the end points are excluded is called an open interval and is denoted by (a, b).
 
Domain and Range of Trigonometric Functions     θ   0    π 2    π    3 π 2    2π      - π 2     - π      -3 π 2   -2π    sin θ   0    1    0     -1     0        -1      0          1     0    cos θ   1     0    -1      0     1        0      - 1          0     1
 
The maximum value of   sin θ  and cos θ , such that 0 ≤ θ ≤ 2π, is 1.
And, the minimum value of sin θ  and cos θ, such that 0 ≤ θ ≤ 2π, is -1 
 
If the value of q is increased or decreased by an integral multiple of 2π, then the values of sine q and cos q do not change.

 -1 ≤ sin θ ≤ 1 and -1 ≤ cos θ ≤ 1, for all θ ∈ R.
 
In other words, the domain of Sine q and Cos q is the set of real numbers, and their range is the closed interval -1 to 1.           Function         Domain         Range          sin θ              R          [-1, 1]          cos θ              R          [-1, 1]          tan θ   R – {(2n+1) π 2 , n ∈ Z}           R
 
tan θ = sin θ cos θ ,  θ ≠  (2n + 1) π 2  , where n is any integer
[ ∵ cos q = 0 ⇒ q = (2n+1) π 2 , where n is any integer]

Or tan q is defined for all q such that it is a real number and is not equal to (2n+1)π/2, where n is any integer. This is the domain of tan q.


And, the range is the set of real numbers R.
 
cot θ =   cos θ sin θ , θ ≠  (2n + 1) π 2 ,  where n is any integer
[∵ sin q = 0 ⇒ q =nπ]
 
Therefore, cot q is defined for all q belonging to the set of real numbers not equal to n π, where n is any integer. This is the domain of cot q and the set of real numbers is the range of cot q.
 
sec θ = 1 cos θ ,  θ ≠  (2n + 1) π 2  , where n is any integer
The value of secant q does not lie between - 1 and 1. Therefore, the range is the union of the closed interval, (-∞, -1), and the closed interval (1, -∞).
In other words, the range is the set of all real numbers y such that y ≥  1 or ≤ -1.
 
cosec θ = 1 sin θ ,θ = nπ,  where n is any integer.         Function          
                  Domain                                                                  Range          cot θ          R – {nπ, n ∈ Z}         R          sec θ         R – {(2n+1)π/2, n ∈ Z} (– ∞, -1]  ∪ [1 , ∞) or, {y: y ∈ R, y ≥ 1 or y ≤ –1}      cosec θ         R – {nπ, n ∈ Z} (– ∞, -1]  ∪ [1 , ∞) or, {y: y ∈ R, y ≥ 1 or y ≤ –1}  
 
Behaviour of trigonometric functions in different quadrants     θ   0    π 2    π    3 π 2    2π      - π 2     - π      -3 π 2   -2π    sin θ   0    1    0     -1     0        -1      0          1     0    cos θ   1     0    -1      0     1        0      - 1          0     1
The value of sin θ increases from 0 to 1 and the value of cos θ decreases from 1 to 0, when θ increases from 0 to π/2.


From π/2 to π, the values of sin θ and cos θ decrease from 1 to 0, and 0 to -1, respectively.


When θ increases from π to 3π/2, sin θ decreases from 0 to -1, while cos θ increases from -1 to 0.


And, in the fourth quadrant, the values of sin θ and cos θ again increase from -1 to 0, and 0 to 1, respectively.
 
The behaviour of the other four functions in different quadrants:
 

 
The values of tan x and cot x repeat after an interval of π.


The values of sin x and cos x repeat after an interval of 2π. Hence, the values of cosec x and sec x will also repeat after an interval of 2π.
 
Graphs of trigonometric functions
 
Sin x     x  0   π 6   π 4   π 3   π 2  π   3 π 2  2π  –  π 2  – π  –  3 π 2  – 2 π  sin x  0   1/2   1/√2   √3/2    1  0    – 1  0    – 1    0      1         0  
sin x = 0, if x = ±π, ±2π, ±3π.....
 
Also, sin x = 1, if x = π 2 , -3 π 2  and sin x = 1, if x =  - π 2 ,  3 π 2 .
 
Sine π/6 is 1/2, sine π /4 is 1/√2, and sine π/3 is √3/2.
 
Plot the values of x on the X-axis and the corresponding values of sin x on the Y-axis to get the graph of the function.
 

 
The curve of sin x passes through the origin, and the minimum and maximum values of sin x are -1 and 1, respectively.
 
The domain is the set of real numbers and the range is the closed interval (-1, 1).
 
Cos x     x  0   π/6     π/4   π/3   π/2    π   3 π/2  2π  –  π/2  – π  –  3 π/2  cos x  0   √3/2   1/√2   1/2    0  – 1     0  1    0    – 1      0     
cos x = 0, if x = ± π 2 , ±  3 π 2  , ±  5 π 2 ,....


Also, cos x = 1, if x = – π, π, and cos x = 1, if x = 2π, – 2π.


Cos π/6 is √3/ 2, cos π/4 is 1/√2 and cos π/3 is ½
 
Plot the values of x on the X-axis and the corresponding values of sin x on the Y-axis to get the graph of the function.
 

The graph of cos x does not pass through the origin, and the minimum and maximum values of cos x are – 1 and 1, respectively.
 
Tan x
 
The values of x are taken as 0, π/6, π/4, π/3, ± π/2, ±π, ±3π/2. The corresponding values of tan x are as shown in the table.     x  0   π/6     π/4   π/3   π/2    π   3 π/2  –  π/2  – π  –  3 π/2  tan x  0    1/√3    1    √3   Not
 Definied  0     Not
Defined   Not
Defined   0     Not
Defined  
Plot the values of x on the X-axis and the corresponding values of sin x on the Y-axis to get the graph of the function.
 

 
The curve of tan x passes through the origin.
 
Cot x       0   π/6     π/4   π/3   π/2    π   3 π/2  2π  –  π/2  – π  –  3 π/2  – 2 π cot x  Not
Defined   √3    1 1/√3    0  Not
Defined    0  Not
Defined   0  Not
Defined      0       Not
Defined  
Like in the other functions, assume the same values of x. Then, the corresponding values of cot x are as shown.
 

 
Potting the ordered pairs (x, cot x) in the Cartesian plane, we get the graph of cot x.
 
Sec x
 
Values of x are taken as 0, π/6, π/4, π/3, ± π/2, ± π, ± 3π/2, ±2 π.
The corresponding values of secant x are as shown.     x  0   π/6     π/4   π/3   π/2    π   3 π/2  2π  –  π/2  – π  –  3 π/2  – 2 π  sec x  0   2/√3   √2   2  Not
Defined  -1  Not
Defined  1  Not
Defined   -1  Not
Defined     1  
Plot the values of x on the X-axis and the corresponding values of sin x on the Y-axis to get the graph of the function.
 

 
The graph of Sec x does not lie between –1 and 1.
 
Cosec x
 
Take the values of x as 0, π/6, π/4, π/3, ± π/2, ± pi, ±3 π/2, ± 2π.
 
The corresponding values of cosecant x are as shown.     x  0   π/6     π/4   π/3  π/2    π  3 π/2   2π  – π/2  – π  – 3 π/2  – 2 π cosec x Not
Defined 2 √2 2/√3 1 Not
Defined    – 1 Not
Defined   -1 Not
Defined      1    Not
Defined
 
Plot the values of x on the X-axis and the corresponding values of sin x on the Y-axis to get the graph of the function.
 

 
The graph of cosec x does not lie between –1 and 1.

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