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Domain, Range, and Graphs of Inverse Trigonometric Functions

Following the above discussions the domain, range, and graphs of inverse trigonometric functions can be summarized as below.
 
In the following figures dotted line graphs are of trigonometric functions and solid line graphs are of corresponding inverse trigonometric functions.
  1. f(x) = sin–1x
     
    Domain: [– 1, 1]
     
    Range (principal values): 112348.png
     
    112342.png
  2. f(x) = cos–1x
     
    Domain: [–1, 1]
     
    Range (principal values): [0, π]
     
    112355.png
  3. f(x) = tan–1x:
     
    Domain: R
     
    Range (principal values): 112370.png
     
    112364.png
  4. f(x) = cot–1x
     
    Domain: R
     
    Range (principal values): (0, π)
     
    112381.png
  5. f(x) = sec–1x
     
    Domain: (–∞, –1] [1, ∞)
     
    Range (principal values): [0, π] – {π/2}
     
    112390.png
  6. F(x) = cosec–1x
     
    Domain: (–∞, –1] [1, ∞)
     
    Range (principal values): [–π/2, π/2] – {0}
     
    112402.png

Some important formulae

  1. i. sin(sin–1x) = x, x [–1, 1]
     
    ii. cos(cos–1x) = x, x [–1, 1]
     
    iii. cosec(cosec–1x) = x, x (–∞, –1] [1, ∞)
     
    iv. sec(sec–1x) = x, x (–∞, –1] [1, ∞)
     
    v. tan(tan–1x) = x, x R
     
    vi. cot(cot–1x) = x, x R
  2. i. sin–1(sin x) = x, x [–π/2, π/2]
     
    ii. cos–1(cos x) = x, x [0, π]
     
    iii. cosec–1(cosec x) = x, x [–π/2, 0) (0, π/2]
     
    iv. sec–1(sec x) = x, x [0, π/2) (π/2, π]
     
    v. tan–1(tan x) = x, x (–π/2, π/2)
     
    vi. cot–1(cot x) = x, x (0, π)
  3. i. sin–1 111633.pngfor all x (–∞, –1] [1, ∞)
     
    ii. cos–1 111627.png for all x (–∞, –1] [1, ∞)
     
    iii. 111621.png
  4. i. 111615.png, for all x [–1, 1]
     
    ii. 111609.png, for all x R
     
    iii. 111603.png for all x (–∞, –1] [1, ∞)
  5. i. 111597.png= 111590.png
     
    ii. 111584.png= 111578.png
     
    iii. 111572.png = 111566.png= 111560.png
  6. i. tan–1 x + tan–1 y111554.png
     
    ii. 111548.png
     
    iii. 111542.pngx ≥ 0, y ≥ 0
     
    iv. cos–1x + cos–1y = 111536.pngx ≥ 0, y ≥ 0
     
    v. cos–1x – cos–1y111530.png
     
    vi. If |x| ≤ 1 then 111524.png
     
     111517.png, x ≥ 0
     
     111511.png, x < 0
     
    vii. If x > 1, then111505.png
     
    viii. If x < –1, then 111499.png
     
    ix. 111493.png
     
    x. 111487.png
     
    xi. 2 cos–1 x = 111481.png
     
    xii. 111475.png
     
    xiii. 111469.png




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