Circular functions, sometimes known as trigonometric functions, can be simply described as functions of a triangle’s angle. This means that the relationship between the angles and sides of a triangle is determined by these trigonometric functions.
The inverse functions of the basic trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant are known as inverse trigonometric functions. Arcus functions, anti trigonometric functions, and cyclometric functions are all names for these functions. In trigonometry, these inverse functions are used to find the angle with any of the trigonometric ratios. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry and navigation.
Let us know about various Trigonometric Functions and their Inverse Properties.
Six Trigonometric Functions
Sine, cosine, and tangent angles are the three primary classes of trigonometric functions. The basic functions can be used to derive the cotangent, secant, and cosecant functions. The other three trigonometric functions are utilized more frequently than the fundamental trigonometric functions. A description of these three key functions can be found in the diagram below. This diagram is known as the sin-cos-tan triangle. Trigonometry is typically defined as a right-angled triangle.
The ratio of the opposite side length to the hypotenuse is known as the sine function of an angle. As illustrated in the diagram, the value of sin is:
CB/CA = Sin a =Opposite/Hypotenuse
The ratio of the neighboring side’s length to the hypotenuse’s length is called as cosine function of an angle. The above diagram can be used to calculate the cos function.
AB/CA = Cos a = Adjacent/Hypotenuse
The tangent function is the ratio of the lengths of the opposing and adjacent sides. It’s worth mentioning that tan can be written as the ratio of sine and cosine. As seen in the picture above, the tan function will be as follows.
CB/BA = CB/BA = Tan a = Opposite/Adjacent
sin a/cos a = tan a
Secant, Cosecant and Cotangent Functions
The three supplementary functions secant, cosecant (csc), and cotangent are derived from the main functions sine, cos, and tan. The reciprocals of sine, cos, and tan are cosecant (csc), secant (sec), and cotangent (cot), respectively. The formula for each of these functions is as follows:
Hypotenuse/Adjacent = CA/AB = Sec a = 1/(cos a)
Concept of Inverse Trigonometric Function
Inverse trigonometric functions, such as sine, cosine, tangent, cosecant, secant, and cotangent, conduct the opposite operation of trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. We already know that trig functions are particularly useful in right-angle triangles. When two sides of a right triangle’s measure are known, these six key functions are employed to obtain the angle measure.
Using arc-prefixes like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), and arccot, the convention symbol denotes the inverse trigonometric function (x).
Angles or real numbers whose sine is x, cosine is x, and tangent is x are denoted by sin-1x, cos-1x, tan-1x, and so on, provided that the answers supplied are numerically the smallest available. These are also known as arcsin x, arccosine x, and so on.
If two angles with the same numerical value, one positive and the other negative, the positive angle should be chosen.
Domain and Range
Many various angles correspond to the same result of sin() in the sine function. As an example,
For any kk integer, We will restrict our domain before calculating the inverse sine function to avoid the problem of numerous values mapping to the same angle.
The original function in the domain indicated above has been flipped around the line y=xy=x, resulting in the graphs of the inverse functions. Because the roles of xx and yy are swapped when the graph is flipped around the line y=xy=x, this observation holds true for the graph of an inverse function.