Inverse Trigonometry Formula

Inverse trigonometric formula is one of the important concepts of Mathematics. It is one of those concepts that lay the basic foundation of a person's understanding of angles and triangles.  Inverse trigonometric functions are closely related to basic trigonometric functions. The domain and range of trigonometric functions are transformed into the domain and range of inverse trigonometric functions. We learn about the relationships between angles and sides in a right-angled triangle in trigonometry. Inverse trigonometry functions are similar. Sin, cos, tan, cosec, sec, and cot are the fundamental trigonometric functions. On the other hand, the inverse trigonometric functions are denoted as sin–1x, cos–1x, cot–1x, tan–1x, cosec–1x, and sec–1x.

Inverse Trigonometric Formulas 

The following formulas have been grouped from the list of inverse trigonometric formulas. These formulas are useful for converting one function to another, determining the functions' principal angle values, and performing a variety of arithmetic operations across these inverse trigonometric functions. Furthermore, all of the basic trigonometric function formulas have been transformed to inverse trigonometric function formulas and are classified as the four sets of formulas listed below.

• Arbitrary Values

• Reciprocal and Complementary functions

• Sum and difference of functions

• Double and triple of a function 

What is an Inverse Trigonometric Function?

In geometry, the part that tells us about the relationships existing between the angles and sides of a right-angled triangle is known as trigonometry. It has formulas and identities that offer great help in mathematical and scientific calculations. As discussed above, trigonometry also has functions and ratios such as sin, cos, and tan. In the same way, we can answer the question of what is an inverse trigonometric function? 

Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. They are also termed as arcus functions, anti trigonometric functions, or cyclometric functions. These inverse functions in trigonometry are used to get the angle with any of the trigonometry ratios. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry, and navigation.

Inverse Trigonometric Functions Formulas

Inverse trigonometric functions formula helps the students to solve the toughest problem easily, all thanks to the inverse trigonometry formula. Some of the inverse trigonometric functions formulas are as follows:

1. sin-1(x) = - sin-1x

2. cos-1(x) = π - cos-1x

3. tan⁻¹(-x) = -tan⁻¹(x)

4. cosec⁻¹(-x) = -cosec⁻¹(x)

5. sec⁻¹(-x) = π - sec⁻¹(x)

6. cot⁻¹(-x) = π - cot⁻¹(x)

7. sin-1(x) + cos-1x = π/2

8. tan-1(x) + cot-1x = π/2

9. sec-1(x) + cosec-1x = π/2

10. tan-1(x)+tan-1(y) = tan-1\[(\frac{x+y}{1-xy})\]

11.  tan-1(x)-tan-1(y) = tan-1\[(\frac{x+y}{1-xy})\]

12. 2sin-1(x) = sin-1(2x\[\sqrt{1-x^2}\])

13. 3sin-1(x) = sin-1(3x - 4x3)

14. sin-1x + sin-1y = sin-1( x\[\sqrt{1-y^2}\] + y\[\sqrt{1-x^2}\]), if x and y ≥ 0 and x2+ y2  ≤ 1

15. cos-1x + cos-1y = cos-1(xy - \[\sqrt{1-x^2}\] + y\[\sqrt{1-y^2}\]), if x and y ≥ 0 and x2 + y2 ≤ 1

So, these were some of the inverse trigonometric functions formulas that you can use while solving trigonometric problems.

Table of Inverse Trigonometric Functions

Inverse Trigonometric Ratios

Inverse trigonometric ratios are trigonometric ratios that are used to find the value of an unknown angle given a value of the right-angled triangle's side ratio. We may use the trigonometric ratios to calculate the angle, just as we used angles to find the trigonometric ratios of the triangle's sides.

Fun Facts

• Hipparchus, the father of trigonometry, compiled the first trigonometry table.

• Inverse trigonometric functions were actually introduced early in the 1700s by Daniel Bernoulli. 

Solved Examples 

Example 1:

Find the value of tan−1(tan 9𝜋/8)

Solution 1:

tan−1(tan 9𝜋/8)

=tan−1(tan(𝜋+𝜋/8))

=tan−1 (tan(𝜋/8))

=𝜋/8

Example 2:

Find sin (cos−1 3/5)

Solution 2:

 Suppose that, (cos−1 3/5) = x

So, cos x = 3/5 

This implies  sin x = cos^-1 \[\frac{3}{5} = \frac{4}{5}\]   

Conclusion

The anti trigonometric functions, also known as arcus functions or cyclometric functions, are inverse trigonometric functions. To get the angle of a triangle using any of the trigonometric functions, use the inverse trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. It is frequently used in a variety of subjects, including geometry, engineering, and physics.