Understanding Inverse Trigonometric Functions: Inverse trigonometric functions, often referred to as arc functions , consist of arcsine, arccosine, arctangent, arc-secant, arc-cotangent, and arc-cosecant. Similar to the standard trigonometric functions , these inverse functions can also be graphically represented.
While sine, cosine, and tangent are used to deduce the length of a right triangle's side, given the length of another side and an acute angle, inverse trigonometric functions help in determining the measure of the angle when the lengths of two sides are known.
Let's delve into the formulas of these functions before exploring their graphical representation.
Formulas for Inverse Trigonometric Functions
Function
Domain
Range of an Inverse Function
sin -1 x or (arcsinex)
-1≤ x ≤1
-π/2 ≤ y ≤ π/2
cos -1 x or (arcosinex)
-1≤ x ≤1
0 ≤ y ≤ π
tan -1 x or (arctangentx)
– ∞ < x < ∞
-π/2 < y < π/2
cot -1 x or (arcotangentx)
– ∞ < x < ∞
0 < y < π
sec -1 x or (arcsecantx)
– ∞ ≤ x ≤-1 or 1≤x≤ ∞
0 ≤ y≤ π, y ≠ π/2
cosec -1 x or (arccosecantx)
– ∞ ≤ x ≤-1 or 1≤x≤ ∞
-π/2 ≤ y≤ π/2, y≠0
It's essential to note the following formulas, considering the domain and range of the inverse functions :
sin( sin -1 x ) = x, if -1 ≤ x ≤ 1 and sin -1 (sin y) = y if -π/2 ≤ y ≤ π/2.
cos(cos -1 x ) = x, if -1 ≤ x ≤ 1 and cos -1 (cos y) = y if 0 ≤ y ≤ π.
tan(tan -1 x ) = x, if -∞ < x < ∞ and cos -1 (cos y) = y if -π/2 ≤ y ≤ π/2.
cot( cot -1 x ) = x, if -∞ < x < ∞ and cot -1 (cot y) = y if 0 < y <π.
sec( sec -1 x ) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ and sec -1 (sec y) = y if 0 ≤ y ≤ π, y ≠ π/2.
cosec( cosec -1 x ) = x, if -∞ ≤ x ≤- 1 or 1 ≤ x ≤ ∞ and cosec -1 (cosec y) = y if -π/2 ≤ y ≤ π/2, y ≠ 0.
Note: The term "Arc Functions" is another name for inverse trigonometric functions. They are so-called because they produce the length of the arc required to achieve a specific value of a trigonometric function.
Graphical Illustration of Inverse Trigonometric Functions
Below, we have provided the graphs of all the inverse trigonometric functions.
Graph of the arcsine function
The arcsine function, denoted by sin -1 x , is the inverse of the sine function. The graph of this function is as follows:
Graph of the arccosine function
The arccosine function, denoted by cos -1 x , is the inverse of the cosine function. Here's its graphical representation:
Graph of the arctangent function
The arctangent function, denoted by tan -1 x , is the inverse of the tangent function. The graph of this function is shown below:
Graph of the arccotangent function
The arccotangent function, denoted by cot -1 x , is the inverse of the cotangent function.
Graph of the arcsecant function
The arcsecant function, denoted by sec -1 x , is the inverse of the secant function. Its graph is shown below:
Graph of the arccosecant function
The arccosecant function, denoted by cosec -1 x , is the inverse of the cosecant function. The graph of this function is given below:
Inverse trigonometric functions are the functions that are also called arc functions. These are arcsine, arccosine, arctangent, arc-secant, arc-cotangent and arc-cosecant. Just like the trigonometric functions, we can also represent graphs of inverse trigonometric functions.
What is the difference between trigonometric functions and inverse trigonometric functions?
Trigonometric functions like sin, cos and tan are used to find the length of the side of the right triangle when we know the length of one side and measure of one of the acute angle. Inverse trigonometric functions, on the other hand, are used to find the measure of the angle when the length of the two sides is known.
What are some applications of the graph of inverse trigonometric functions?
Graphs of inverse trigonometric functions are used in solving problems in trigonometry. They are particularly useful in determining the length of arc needed to obtain a particular value of a trigonometric function.