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Properties of Inverse Trigonometric Functions
Properties of Inverse Trigonometric Functions The inverse trigonometric functions provide the unique angle whose trigonometric ratio is known. These function...
Properties of Inverse Trigonometric Functions The inverse trigonometric functions provide the unique angle whose trigonometric ratio is known. These function...
The inverse trigonometric functions provide the unique angle whose trigonometric ratio is known. These functions allow us to find the angle from which the trigonometric ratio is known.
Here are some key properties of the inverse trigonometric functions:
Symmetry: All inverse trigonometric functions are symmetric with respect to the quadrant in which they are defined. This means that f^(-x) = f(x).
Identity property: For any angle θ, we have:
θ = arccos(sin(θ)) = arccos(cos(θ))
tan(θ1/2) = sin(θ1)/cos(θ1) = sin(θ2)/cos(θ2)
Range: The range of each inverse trigonometric function is the interval [-1, 1].
Domain: The domain of each inverse trigonometric function is the interval [0, π].
One-to-one property: Each inverse trigonometric function is one-to-one, meaning that each unique angle has exactly one corresponding trigonometric ratio.
Composition: The composition of two inverse trigonometric functions is also an inverse trigonometric function.
Examples:
Arccos(0.5) = π/3 (in the first quadrant)
arctan(√3) = π/3 (in the first quadrant)
arccot(-1) = π + π/2 (in the third quadrant)
These properties allow us to perform various operations with inverse trigonometric functions, such as finding angles from known trigonometric ratios, finding the angle whose trigonometric ratio is known, and composing functions that involve inverse trigonometric functions