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INVERSE TRIGONOMETRIC FUNCTIONS CLASS XII

INVERSE TRIGONOMETRIC FUNCTIONS CLASS XII. y. x. y = sin x. Inverse Sine Function. Inverse Sine Function. Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. f ( x ) = sin x does not pass the Horizontal Line Test.

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INVERSE TRIGONOMETRIC FUNCTIONS CLASS XII

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  1. INVERSE TRIGONOMETRIC FUNCTIONSCLASS XII

  2. y x y = sin x Inverse Sine Function Inverse Sine Function Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. f(x) = sin x does not pass the Horizontal Line Test and must be restricted to find its inverse. Sin x has an inverse function on this interval. Created By ----- S P Dwivedi

  3. Angle whose sine is x This is another way to write arcsin x. Inverse Sine Function The inverse sine function is defined by y = arcsin x if and only if sin y = x. The domain of y = arcsinxis [–1, 1]. The range of y = arcsin x is [–/2 , /2]. Example: Created By ----- S P Dwivedi

  4. y x y = cos x Inverse Cosine Function Inverse Cosine Function f(x) = cos x must be restricted to find its inverse. Cos x has an inverse function on this interval. Created By ----- S P Dwivedi

  5. Angle whose cosine is x This is another way to write arccos x. Inverse Cosine Function The inverse cosine function is defined by y = arccos x if and only if cos y = x. The domain of y = arccos x is [–1, 1]. The range of y = arccos x is [0 , ]. Example: Created By ----- S P Dwivedi

  6. y = tan x y x Inverse Tangent Function Inverse Tangent Function f(x) = tan x must be restricted to find its inverse. Tan x has an inverse function on this interval. Created By ----- S P Dwivedi

  7. Angle whose tangent is x The domain of y = arctan x is . This is another way to write arctan x. Inverse Tangent Function The inverse tangent function is defined by y = arctan x if and only if tan y = x. The range of y = arctan x is [–/2 , /2]. Example: Created By ----- S P Dwivedi

  8. Composition of Functions Composition of Functions: f(f –1(x)) = x and (f –1(f(x)) = x. Inverse Properties: If –1  x  1 and – /2  y  /2, then sin(arcsin x) = x and arcsin(sin y) = y. If –1  x  1 and 0  y  , then cos(arccos x) = x and arccos(cos y) = y. If x is a real number and –/2 < y < /2, then tan(arctan x) = x and arctan(tan y) = y. Example: tan(arctan 4) = 4 Created By ----- S P Dwivedi

  9. does not lie in the range of the arcsine function, –/2  y  /2. y However, it is coterminal with which does lie in the range of the arcsine function. x Composition of Functions Example: a. sin–1(sin (–/2)) = –/2 Created By ----- S P Dwivedi

  10. y x u Example: Evaluating Composition of Functions Example: 3 2 Created By ----- S P Dwivedi

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