There are particularly six inverse trig functions for each trigonometry ratio. The inverse of six important trigonometric functions are:
- Arcsine
- Arccosine
- Arctangent
- Arccotangent
- Arcsecant
- Arccosecant
Let us discuss all the six important types of inverse trigonometric functions along with its definition, formulas, graphs, properties and solved examples.
Arcsine Function
Arcsine function is an inverse of the sine function denoted by sin-1x. It is represented in the graph as shown below:
| Domain | -1 ≤ x ≤ 1 |
| Range | -π/2 ≤ y ≤ π/2 |
Arccosine Function
Arccosine function is the inverse of the cosine function denoted by cos-1x. It is represented in the graph as shown below:
Therefore, the inverse of cos function can be expressed as; y = cos-1x (arccosine x)
Domain & Range of arcsine function:
| Domain | -1≤x≤1 |
| Range | 0 ≤ y ≤ π |
Arctangent Function
Arctangent function is the inverse of the tangent function denoted by tan-1x. It is represented in the graph as shown below:
Therefore, the inverse of tangent function can be expressed as; y = tan-1x (arctangent x)
Domain & Range of Arctangent:
| Domain | -∞ < x < ∞ |
| Range | -π/2 < y < π/2 |
Arccotangent (Arccot) Function
Arccotangent function is the inverse of the cotangent function denoted by cot-1x. It is represented in the graph as shown below:
Therefore, the inverse of cotangent function can be expressed as; y = cot-1x (arccotangent x)
Domain & Range of Arccotangent:
| Domain | -∞ < x < ∞ |
| Range | 0 < y < π |
Arcsecant Function
What is arcsecant (arcsec)function? Arcsecant function is the inverse of the secant function denoted by sec-1x. It is represented in the graph as shown below:
Therefore, the inverse of secant function can be expressed as; y = sec-1x (arcsecant x)
Domain & Range of Arcsecant:
| Domain | -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ |
| Range | 0 ≤ y ≤ π, y ≠ π/2 |
Arccosecant Function
What is arccosecant (arccsc x) function? Arccosecant function is the inverse of the cosecant function denoted by cosec-1x. It is represented in the graph as shown below:
Therefore, the inverse of cosecant function can be expressed as; y = cosec-1x (arccosecant x)
Domain & Range of Arccosecant is:
| Domain | -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ |
| Range | -π/2 ≤ y ≤ π/2, y ≠ 0 |
Inverse Trigonometric Functions Table
Let us rewrite here all the inverse trigonometric functions with their notation, definition, domain and range.
| 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 ≤ π0° ≤ y ≤ 180° |
| Arctangent orInverse tangent | y=tan-1(x) | x=tan y | For all real numbers | − π/2 < y < π/2-90°< y < 90° |
| Arccotangent orInverse Cot | y=cot-1(x) | x=cot y | For all real numbers | 0 < y < π0° < y < 180° |
| Arcsecant orInverse Secant | y = sec-1(x) | x=sec y | x ≤ −1 or 1 ≤ x | 0≤y<π/2 or π/2<y≤π0°≤y<90° or 90°<y≤180° |
| Arccosecant | 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° |
Leave a Reply