Inverse Trigonometric Functions - Formulas, Graphs & Problems
Inverse trigonometric functions are mathematical functions that calculate the angle or angle measure given the ratio of two sides of a right-angled triangle. In other words, they are the inverse operations of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent).
For example, if we know the value of sine of an angle, we can use the arcsine function (or sin-1 function) to find the angle. Similarly, if we know the value of cosine of an angle, we can use the arccosine function (or cos-1 function) to find the angle, and so on for the other trigonometric functions.
Inverse Trigonometric Formulas
here are the basic inverse trigonometric formulas:
| Function | Notation | Formula | Domain |
|---|---|---|---|
| Arcsine | sin-1(x) or arcsin(x) | sin-1(-x) = -sin-1(x) | x ∈ [-1, 1] |
| Arccosine | cos-1(x) or arccos(x) | cos-1(-x) = π - cos-1(x) | x ∈ [-1, 1] |
| Arctangent | tan-1(x) or arctan(x) | tan-1(-x) = -tan-1(x) | x ∈ R |
| Arccotangent | cot-1(x) or arccot(x) | cot-1(-x) = π - cot-1(x) | x ∈ R |
| Arcsecant | sec-1(x) or arcsec(x) | sec-1(-x) = π - sec-1(x) | |x| ≥ 1 |
| Arccosecant | csc-1(x) or arccsc(x) | csc-1(-x) = -csc-1(x) | |x| ≥ 1 |
Inverse Trigonometric Functions Table
| Function Name | Notation | Definition | Domain of x | Range |
|---|---|---|---|---|
| Arcsine or inverse sine | y = sin-1(x) | x = sin y | -1 ≤ x ≤ 1 | -π/2 ≤ y ≤ π/2 -90° ≤ y ≤ 90° |
| Arccosine or inverse cosine | y = cos-1(x) | x = cos y | -1 ≤ x ≤ 1 | 0 ≤ y ≤ π 0° ≤ y ≤ 180° |
| Arctangent or inverse tangent | y = tan-1(x) | x = tan y | For all real numbers | -π/2 < y < π/2 -90° < y < 90° |
| Arccotangent or inverse cot | y = cot-1(x) | x = cot y | For all real numbers | 0 < y < π 0° < y < 180° |
| Arcsecant or inverse secant | y = sec-1(x) | x = sec y | x ≤ -1 or 1 ≤ x | 0 ≤ y < π/2 or π/2 < y ≤ π 0° ≤ y < 90° or 90° < y ≤ 180° |
| Arccosecant or inverse cosecant | y = csc-1(x) | x = csc y | x ≤ -1 or 1 ≤ x | -π/2 ≤ y < 0 or 0 < y ≤ π/2 -90° ≤ y < 0° or 0° < y ≤ 90° |
Inverse Trigonometric Functions Derivatives
| Inverse Trig Function | dy/dx |
|---|---|
| y = sin-1(x) | 1/√(1-x2) |
| y = cos-1(x) | -1/√(1-x2) |
| y = tan-1(x) | 1/(1+x2) |
| y = cot-1(x) | -1/(1+x2) |
| y = sec-1(x) | 1/[|x|√(x2-1)] |
| y = csc-1(x) | -1/[|x|√(x2-1)] |
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