Notes On Domain and Range of Trigonometric Functions - CBSE 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     θ   0   $\frac{\pi }{\text{2}}$    π   $\frac{\text{3}\pi }{\text{2}}$    2π     $\frac{\text{-}\pi }{\text{2}}$     - π     $\frac{\text{-3}\pi }{\text{2}}$   -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  and cos θ , such that 0 ≤ θ ≤ 2π, is 1. And, the minimum value of 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)$\frac{\pi }{\text{2}}$, n ∈ Z}           R   tan θ = $\frac{sin\theta \text{}}{cos\theta \text{}}\text{,}$ θ ≠ $\frac{\text{(2n + 1)}\pi }{\text{2}}$ , where n is any integer [$\because \text{}$cos q = 0 ⇒ q = (2n+1)$\frac{\pi }{\text{2}}\text{}$, 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)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 =  $\frac{cos\theta \text{}}{sin\theta \text{}}$, θ ≠ $\frac{\text{(2n + 1)}\pi }{\text{2}}$,  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 = $\frac{\text{1}}{cos\theta \text{}}$,  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 = $\frac{\text{1}}{sin\theta \text{}}$,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  $\frac{\pi }{\text{6}}$  $\frac{\pi }{\text{4}}$  $\frac{\pi }{\text{3}}$  $\frac{\pi }{\text{2}}$  π  $\frac{\text{3}\pi }{\text{2}}$  2π  – $\frac{\pi }{\text{2}}$  – π  – $\frac{\text{3}\pi }{\text{2}}$  – 2$\pi$  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 = $\frac{\pi }{\text{2}}$, $\frac{\text{-3}\pi }{\text{2}}$ and sin x = 1, if x = $\frac{\text{-}\pi }{\text{2}}$ , $\frac{\text{3}\pi }{\text{2}}$ .   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 = ±$\frac{\text{}\pi }{\text{2}}$, ± $\frac{\text{}\text{3}\pi }{\text{2}}$ , ± $\frac{\text{5}\pi }{\text{2}}$,.... 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     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.

#### 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   $\frac{\pi }{\text{2}}$    π   $\frac{\text{3}\pi }{\text{2}}$    2π     $\frac{\text{-}\pi }{\text{2}}$     - π     $\frac{\text{-3}\pi }{\text{2}}$   -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  and cos θ , such that 0 ≤ θ ≤ 2π, is 1. And, the minimum value of 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)$\frac{\pi }{\text{2}}$, n ∈ Z}           R   tan θ = $\frac{sin\theta \text{}}{cos\theta \text{}}\text{,}$ θ ≠ $\frac{\text{(2n + 1)}\pi }{\text{2}}$ , where n is any integer [$\because \text{}$cos q = 0 ⇒ q = (2n+1)$\frac{\pi }{\text{2}}\text{}$, 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)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 =  $\frac{cos\theta \text{}}{sin\theta \text{}}$, θ ≠ $\frac{\text{(2n + 1)}\pi }{\text{2}}$,  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 = $\frac{\text{1}}{cos\theta \text{}}$,  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 = $\frac{\text{1}}{sin\theta \text{}}$,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  $\frac{\pi }{\text{6}}$  $\frac{\pi }{\text{4}}$  $\frac{\pi }{\text{3}}$  $\frac{\pi }{\text{2}}$  π  $\frac{\text{3}\pi }{\text{2}}$  2π  – $\frac{\pi }{\text{2}}$  – π  – $\frac{\text{3}\pi }{\text{2}}$  – 2$\pi$  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 = $\frac{\pi }{\text{2}}$, $\frac{\text{-3}\pi }{\text{2}}$ and sin x = 1, if x = $\frac{\text{-}\pi }{\text{2}}$ , $\frac{\text{3}\pi }{\text{2}}$ .   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 = ±$\frac{\text{}\pi }{\text{2}}$, ± $\frac{\text{}\text{3}\pi }{\text{2}}$ , ± $\frac{\text{5}\pi }{\text{2}}$,.... 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     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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