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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     Î¸   0    Ï€ 2    Ï€    3 Ï€ 2    2Ï€      - Ï€ 2     - Ï€      -3 Ï€ 2   -2Ï€    sin Î¸   0    1    0     -1     0        -1      0          1     0    cos Î¸   1     0    -1      0     1        0      - 1          0     1

The maximum value of   sin Î¸  and cos Î¸ , such that 0 â‰¤ Î¸ â‰¤ 2Ï€, is 1.
And, the minimum value of sin Î¸  and cos Î¸, such that 0 â‰¤ Î¸ â‰¤ 2Ï€, is -1

If the value of q is increased or decreased by an integral multiple of 2Ï€, then the values of sine q and cos q do not change.

â‡’ -1 â‰¤ sin Î¸ â‰¤ 1 and -1 â‰¤ cos Î¸ â‰¤ 1, for all Î¸ âˆˆ 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 Î¸              R          [-1, 1]          cos Î¸              R          [-1, 1]          tan Î¸   R â€“ {(2n+1) Ï€ 2 , n âˆˆ Z}           R

tan Î¸ = sin Î¸ cos Î¸ ,  Î¸ â‰   (2n + 1) Ï€ 2  , where n is any integer
[ âˆµ cos q = 0 â‡’ q = (2n+1) Ï€ 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)Ï€/2, where n is any integer. This is the domain of tan q.

And, the range is the set of real numbers R.

cot Î¸ =   cos Î¸ sin Î¸ , Î¸ â‰   (2n + 1) Ï€ 2 ,  where n is any integer
[âˆµ sin q = 0 â‡’ q =nÏ€]

Therefore, cot q is defined for all q belonging to the set of real numbers not equal to n Ï€, 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 Î¸ = 1 cos Î¸ ,  Î¸ â‰   (2n + 1) Ï€ 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, (-âˆž, -1), and the closed interval (1, -âˆž).
In other words, the range is the set of all real numbers y such that y â‰¥  1 or â‰¤ -1.

cosec Î¸ = 1 sin Î¸ ,Î¸ = nÏ€,  where n is any integer.         Function
Domain                                                                  Range          cot Î¸          R â€“ {nÏ€, n âˆˆ Z}         R          sec Î¸         R â€“ {(2n+1)Ï€/2, n âˆˆ Z} (â€“ âˆž, -1]  âˆª [1 , âˆž) or, {y: y âˆˆ R, y â‰¥ 1 or y â‰¤ â€“1}      cosec Î¸         R â€“ {nÏ€, n âˆˆ Z} (â€“ âˆž, -1]  âˆª [1 , âˆž) or, {y: y âˆˆ R, y â‰¥ 1 or y â‰¤ â€“1}

Behaviour of trigonometric functions in different quadrants     Î¸   0    Ï€ 2    Ï€    3 Ï€ 2    2Ï€      - Ï€ 2     - Ï€      -3 Ï€ 2   -2Ï€    sin Î¸   0    1    0     -1     0        -1      0          1     0    cos Î¸   1     0    -1      0     1        0      - 1          0     1
The value of sin Î¸ increases from 0 to 1 and the value of cos Î¸ decreases from 1 to 0, when Î¸ increases from 0 to Ï€/2.

From Ï€/2 to Ï€, the values of sin Î¸ and cos Î¸ decrease from 1 to 0, and 0 to -1, respectively.

When Î¸ increases from Ï€ to 3Ï€/2, sin Î¸ decreases from 0 to -1, while cos Î¸ increases from -1 to 0.

And, in the fourth quadrant, the values of sin Î¸ and cos Î¸ 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 Ï€.

The values of sin x and cos x repeat after an interval of 2Ï€. Hence, the values of cosec x and sec x will also repeat after an interval of 2Ï€.

Graphs of trigonometric functions

Sin x     x  0   Ï€ 6   Ï€ 4   Ï€ 3   Ï€ 2  Ï€   3 Ï€ 2  2Ï€  â€“  Ï€ 2  â€“ Ï€  â€“  3 Ï€ 2  â€“ 2 Ï€  sin x  0   1/2   1/âˆš2   âˆš3/2    1  0    â€“ 1  0    â€“ 1    0      1         0
sin x = 0, if x = ±Ï€, ±2Ï€, ±3Ï€.....

Also, sin x = 1, if x = Ï€ 2 , -3 Ï€ 2  and sin x = 1, if x =  - Ï€ 2 ,  3 Ï€ 2 .

Sine Ï€/6 is 1/2, sine Ï€ /4 is 1/âˆš2, and sine Ï€/3 is âˆš3/2.

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   Ï€/6     Ï€/4   Ï€/3   Ï€/2    Ï€   3 Ï€/2  2Ï€  â€“  Ï€/2  â€“ Ï€  â€“  3 Ï€/2  cos x  0   âˆš3/2   1/âˆš2   1/2    0  â€“ 1     0  1    0    â€“ 1      0
cos x = 0, if x = ± Ï€ 2 , ±  3 Ï€ 2  , ±  5 Ï€ 2 ,....

Also, cos x = 1, if x = â€“ Ï€, Ï€, and cos x = 1, if x = 2Ï€, â€“ 2Ï€.

Cos Ï€/6 is âˆš3/ 2, cos Ï€/4 is 1/âˆš2 and cos Ï€/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 â€“ 1 and 1, respectively.

Tan x

The values of x are taken as 0, Ï€/6, Ï€/4, Ï€/3, ± Ï€/2, ±Ï€, ±3Ï€/2. The corresponding values of tan x are as shown in the table.     x  0   Ï€/6     Ï€/4   Ï€/3   Ï€/2    Ï€   3 Ï€/2  â€“  Ï€/2  â€“ Ï€  â€“  3 Ï€/2  tan x  0    1/âˆš3    1    âˆš3   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   Ï€/6     Ï€/4   Ï€/3   Ï€/2    Ï€   3 Ï€/2  2Ï€  â€“  Ï€/2  â€“ Ï€  â€“  3 Ï€/2  â€“ 2 Ï€ cot x  Not
Defined   âˆš3    1 1/âˆš3    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, Ï€/6, Ï€/4, Ï€/3, ± Ï€/2, ± Ï€, ± 3Ï€/2, ±2 Ï€.
The corresponding values of secant x are as shown.     x  0   Ï€/6     Ï€/4   Ï€/3   Ï€/2    Ï€   3 Ï€/2  2Ï€  â€“  Ï€/2  â€“ Ï€  â€“  3 Ï€/2  â€“ 2 Ï€  sec x  0   2/âˆš3   âˆš2   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 â€“1 and 1.

Cosec x

Take the values of x as 0, Ï€/6, Ï€/4, Ï€/3, ± Ï€/2, ± pi, ±3 Ï€/2, ± 2Ï€.

The corresponding values of cosecant x are as shown.     x  0   Ï€/6     Ï€/4   Ï€/3  Ï€/2    Ï€  3 Ï€/2   2Ï€  â€“ Ï€/2  â€“ Ï€  â€“ 3 Ï€/2  â€“ 2 Ï€ cosec x Not
Defined 2 âˆš2 2/âˆš3 1 Not
Defined    â€“ 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 â€“1 and 1.