Notes On Domain and Range of Trigonometric Functions - Gujarat board Class 11 Maths
  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   $\frac{Ï€}{\text{2}}$    Ï€   $\frac{\text{3}Ï€}{\text{2}}$    2Ï€     $\frac{\text{-}Ï€}{\text{2}}$     - Ï€     $\frac{\text{-3}Ï€}{\text{2}}$   -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  and cos Î¸ , such that 0 â‰¤ Î¸ â‰¤ 2Ï€, is 1. And, the minimum value of 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)$\frac{Ï€}{\text{2}}$, n âˆˆ Z}           R   tan Î¸ = $\frac{sin\mathrm{Î¸}\text{}}{cos\mathrm{Î¸}\text{}}\text{,}$ Î¸ â‰  $\frac{\text{(2n + 1)}Ï€}{\text{2}}$ , where n is any integer [$âˆµ\text{}$cos q = 0 â‡’ q = (2n+1)$\frac{Ï€}{\text{2}}\text{}$, 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¸ =  $\frac{cos\mathrm{Î¸}\text{}}{sin\mathrm{Î¸}\text{}}$, Î¸ â‰  $\frac{\text{(2n + 1)}Ï€}{\text{2}}$,  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¸ = $\frac{\text{1}}{cos\mathrm{Î¸}\text{}}$,  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¸ = $\frac{\text{1}}{sin\mathrm{Î¸}\text{}}$,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  $\frac{Ï€}{\text{6}}$  $\frac{Ï€}{\text{4}}$  $\frac{Ï€}{\text{3}}$  $\frac{Ï€}{\text{2}}$  Ï€  $\frac{\text{3}Ï€}{\text{2}}$  2Ï€  â€“ $\frac{Ï€}{\text{2}}$  â€“ Ï€  â€“ $\frac{\text{3}Ï€}{\text{2}}$  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 = Â±Ï€, Â±2Ï€, Â±3Ï€.....   Also, sin x = 1, if x = $\frac{Ï€}{\text{2}}$, $\frac{\text{-3}Ï€}{\text{2}}$ and sin x = 1, if x = $\frac{\text{-}Ï€}{\text{2}}$ , $\frac{\text{3}Ï€}{\text{2}}$ .   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 = Â±$\frac{\text{}Ï€}{\text{2}}$, Â± $\frac{\text{}\text{3}Ï€}{\text{2}}$ , Â± $\frac{\text{5}Ï€}{\text{2}}$,.... 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 Â½   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, Â± Ï€/2, Â±Ï€, Â±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     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, Â± Ï€/2, Â± Ï€, Â± 3Ï€/2, Â±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, Â± Ï€/2, Â± pi, Â±3 Ï€/2, Â± 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.

#### 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   $\frac{Ï€}{\text{2}}$    Ï€   $\frac{\text{3}Ï€}{\text{2}}$    2Ï€     $\frac{\text{-}Ï€}{\text{2}}$     - Ï€     $\frac{\text{-3}Ï€}{\text{2}}$   -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  and cos Î¸ , such that 0 â‰¤ Î¸ â‰¤ 2Ï€, is 1. And, the minimum value of 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)$\frac{Ï€}{\text{2}}$, n âˆˆ Z}           R   tan Î¸ = $\frac{sin\mathrm{Î¸}\text{}}{cos\mathrm{Î¸}\text{}}\text{,}$ Î¸ â‰  $\frac{\text{(2n + 1)}Ï€}{\text{2}}$ , where n is any integer [$âˆµ\text{}$cos q = 0 â‡’ q = (2n+1)$\frac{Ï€}{\text{2}}\text{}$, 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¸ =  $\frac{cos\mathrm{Î¸}\text{}}{sin\mathrm{Î¸}\text{}}$, Î¸ â‰  $\frac{\text{(2n + 1)}Ï€}{\text{2}}$,  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¸ = $\frac{\text{1}}{cos\mathrm{Î¸}\text{}}$,  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¸ = $\frac{\text{1}}{sin\mathrm{Î¸}\text{}}$,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  $\frac{Ï€}{\text{6}}$  $\frac{Ï€}{\text{4}}$  $\frac{Ï€}{\text{3}}$  $\frac{Ï€}{\text{2}}$  Ï€  $\frac{\text{3}Ï€}{\text{2}}$  2Ï€  â€“ $\frac{Ï€}{\text{2}}$  â€“ Ï€  â€“ $\frac{\text{3}Ï€}{\text{2}}$  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 = Â±Ï€, Â±2Ï€, Â±3Ï€.....   Also, sin x = 1, if x = $\frac{Ï€}{\text{2}}$, $\frac{\text{-3}Ï€}{\text{2}}$ and sin x = 1, if x = $\frac{\text{-}Ï€}{\text{2}}$ , $\frac{\text{3}Ï€}{\text{2}}$ .   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 = Â±$\frac{\text{}Ï€}{\text{2}}$, Â± $\frac{\text{}\text{3}Ï€}{\text{2}}$ , Â± $\frac{\text{5}Ï€}{\text{2}}$,.... 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 Â½   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, Â± Ï€/2, Â±Ï€, Â±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     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, Â± Ï€/2, Â± Ï€, Â± 3Ï€/2, Â±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, Â± Ï€/2, Â± pi, Â±3 Ï€/2, Â± 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.

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