Class 12 Maths Inverse Trigonometric Functions Trigonometric Inverse

Trigonometric Inverse

Inverse trigonometric functions are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

If we are given that the value of the sine function is 1/7, then the we have to find the radian angle x. sin x = 1/7, so x = sin-1 1/7

For inverse to exist, function must be 1:1 , onto

Trigonometric functions are neither 1:1, nor onto over their natural domains and ranges. Eg y= sin x is not one-one & onto over its natural range & domain.

To make these trigonometric functions one-one & onto, we restrict domains & ranges of these trigonometric functions to ensure existence of their inverses.

Natural domain & range of trigonometric functions

• sine function, i.e., sine : R→ [– 1, 1]
• cosine function, i.e., cos : R → [– 1, 1]
• tangent function, i.e., tan : R – { x : x = (2n + 1) π/2, n Z} → R
• cotangent function, i.e., cot : R – { x : x = nπ, n Z} → R
• secant function, i.e., sec : R – { x : x = (2n + 1) π/2, n Z} → R – (– 1, 1)
• cosecant function, i.e., cosec : R – { x : x = nπ, n Z} → R – (– 1, 1)

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