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Introduction to Inverse Trigonometric Functions

In Mathematics, the trigonometric function is a function which defines the relationship between the angles and the sides of a right triangle. As we know, a right triangle is a triangle in which one of the angles should be of 90 degrees. The side adjacent to the angle is called the base or adjacent side, and the side opposite to the angle is called the perpendicular, and the remaining side is called the hypotenuse, which is the longest side of a right triangle. Based on the sides and the angle, the six trigonometric functions formed are sine, cosine, tangent, cosecant, secant and the cotangent.

In arithmetic, if there is an operation, there is a way to calculate the opposite of the operation. For instance, the opposite of addition operation is subtraction, the opposite of multiplication is division, logarithmic functions are the inverse function of an exponential function, and so on. Likewise, the inverse of trigonometric functions is the inverse trigonometric functions.

Among these six trigonometric functions mentioned above, the three primary functions are sine (sin), cosine (cos), and tangent (tan). The other three functions, such as cosecant (csc), secant (sec), and the cotangent (cot) are the inverse/reciprocal of sin, cos and tan respectively. The inverse of these trigonometric functions is called inverse trigonometric functions. It is also called the “Arc Functions”. The six important inverse trig functions are:

  • Arcsine (or) arcsin (or) asin
  • Arccosine (or) arccos (or) acos
  • Arctangent (or) arctan (or)atan
  • Arccosecant (or) arccosec (or) acosec
  • Arcsecant (or) arcsec (or) asec
  • Arccotangent (or)arccot (or) acot

Now, let us consider an example to find the solutions for a few trigonometric and inverse trigonometric functions.

Assume a right triangle, whose measures are:

Opposite side = 3 cm

Adjacent side = 4 cm

Hypotenuse = 5 cm

Finding Sine and Sine Inverse:

We know that, sine θ = Opposite side/ Hypotenuse = 3/5 = 0.6

Thus, the sine function for the given data is 0.6.

In the case of finding the value of θ, we should use the sine inverse function.

It means that

Θ = sin-1 (opposite side/hypotenuse)

Θ = Sin-1 (0.6)

Θ = 36.86°.

As we know, the sine function is the ratio of the opposite side to the hypotenuse. It means that the sine function will take an angle, and it results in the ratio “opposite side/hypotenuse”. But the inverse sine (i.e., sin-1) will take the ratio of the length of the opposite side to the hypotenuse, and it gives an angle. 

Similarly,  we can find the values of other inverse trigonometric functions.

Cos θ = Adjacent side / Hypotenuse 

⇒θ = cos-1 (Adjacent side/ Hypotenuse) 

⇒θ = cos-1(4/5) = 36.86°

Tan θ = Opposite side / Adjacent side

⇒θ = tan-1 (opposite side/ adjacent side)

⇒θ = tan-1(3/4) = 36.86°

Hence, we can say that the value of θ is 36.86°. 

Thus, the inverse trigonometric function helps to find the angle, if we know all the side length of a right triangle. It also has a wide range of applications in various fields like Physics, Navigation, Engineering, and so on.

 

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