Inverse Trigonometric Functions and Their Properties Inverse trigonometric functions are the functions that undo the trigonometric functions. For example...
Inverse trigonometric functions are the functions that undo the trigonometric functions. For example, if (y = \sin(x)), then (x = \arcsin(y)). This means that the inverse trigonometric function tells us the angle whose sine is equal to (y).
Properties of Inverse Trigonometric Functions:
Inverse sine function:
(\arcsin(y)) is the angle whose sine is (y).
(\sin^{-1}(y)) is the angle whose sine is equal to (y).
For any angle (x), (\arcsin(x)) is the angle in the first quadrant whose sine is equal to (x).
Inverse cosine function:
(\arccos(y)) is the angle whose cosine is (y).
(\cos^{-1}(y)) is the angle whose cosine is equal to (y).
For any angle (x), (\arccos(x)) is the angle in the first quadrant whose cosine is equal to (x).
Inverse tangent function:
(\arctan(y)) is the angle whose tangent is (y).
(\tan^{-1}(y)) is the angle whose tangent is equal to (y).
For any angle (x), (\arctan(x)) is the angle in the first quadrant whose tangent is equal to (x).
Inverse cotangent function:
(\arccot(y)) is the angle whose cotangent is (y).
(\cot^{-1}(y)) is the angle whose cotangent is equal to (y).
For any angle (x), (\arccot(x)) is the angle in the first quadrant whose cotangent is equal to (x).
These properties allow us to find the angle that corresponds to a given trigonometric value, and vice versa. For example, if we know that (\sin(x) = \frac{1}{2}), then (x = \arcsin(\frac{1}{2}) = \frac{\pi}{6})