Introduction to Inverse Trigonometric Functions

In our article on inverse trigonometric functions, we will discuss the basic concepts of inverse trigonometric functions, inverse trigonometric identities, and inverse trigonometric functions problems and solutions. 

Basic concepts of inverse trigonometric functions 

• The branch of mathematics that deals with the angles and sides is called trigonometry. 
• The concept of inverse trigonometry deals with the inverse functions of the trigonometric functions. Therefore, the inverse trigonometric functions are inverse cotangent, inverse cosecant, inverse sine, inverse tangent, inverse secant, and inverse cosine. 
• When only two sides of a right triangle are known, the inverse trigonometric functions determine the angle measure.  
• The concept of inverse trigonometric functions is generally used in physics, geometry, engineering, etc. 
• Inverse trigonometric functions are also known as anti-trigonometric functions or arcus functions. 

Properties of inverse trigonometric functions

The following is a list of inverse trigonometric identities and inverse trigonometry formulas. 

• The first property of inverse trigonometric functions-

• sin-1  1x = cosec-1 x, provided x is either greater than or equal to and less than or equal to -1. 

• cos-1  1x = sec-1 x, provided x is either greater than or equal to and less than or equal -1. 

• tan -1  1x = = cot-1 x, provided x is either greater than zero. 

Now, let’s prove the first property. 

Let sec-1 x = y.

Therefore, x = sec y, 

              1x = cos y 

Hence, cos-1  1x = y

Or, cos-1  1x = sec-1 x 

• The second property of inverse trigonometric functions

1. sin-1(-x) = – sin-1 x, for all values of x belonging in a range of -1 to 1. 
2. tan-1(-x) = – tan-1 x, where x ∈ R. 
3. cosec-1(-x) = – cosec-1 x, x 1

Now, let’s prove the second property with the help of an example. 

Let tan-1(-x) = y…. (1)

Then, (-x) = tan y 

Therefore, x = – tan y 

x = tan (-y) 

tan-1 x = (-y) = {replace the value of y from equation 1)

tan-1 x = – tan-1(-x)

• Third property of inverse trigonometric functions 

1. cos-1(-x) = – cos-1 x, where x belongs within a range of -1 to 1. 
2. sec-1(-x) = – sec-1 x, x 1. 
3. cot-1(-x) = – cot-1 x, where x ∈ R.

Now let’s prove the third property.

Let cot–1 (–x) = y

– x = cot y 

so that x = – cot y = cot (π – y) 

Therefore, cot–1 x = π – y = π – cot–1 (–x) 

Hence cot–1 (–x) = π – cot–1 x

• The fourth property of inverse trigonometric functions

1. sin-1 x + cos-1 x = 2, for all x belonging within the range of -1 to 1. 
2. tan-1 x + cot-1 x = 2, where x ∈ R.
3. cosec-1 x + sec-1 x = 2, x 1.

Now, let’s prove the fourth property. 

Let tan-1 x = y. 

Then, x = cot y 

X = cot (2 – y) 

 cot-1x = 2 – y =  2 – tan-1x

Therefore, tan-1 x + cot-1 x = 2

• Fifth property of inverse trigonometric functions 

1. tan-1 x + tan-1 y = tan-1x+y1-xy , if xy < 1.
2. tan-1 x – tan-1 y = tan-1x-y1+xy , if xy > -1. 
3. tan-1 x + tan-1 y = + tan-1x+y1-xy , xy > 1; x, y>0. 

• Sixth property of inverse trigonometric functions 

1. 2tan-1 x = sin-12x/1+x2, x 1. 
2. 2tan-1 x = cos-11-x21+ x2, x 0. 
3. 2tan-1 x = tan-1 2×1- x2, if x is either greater than -1 or less than 1. 

While there are only six properties of inverse trigonometric functions, there are still some inverse trigonometric identities and inverse trigonometry formulas that are left uncovered. Therefore, the following list has some more inverse trigonometric identities- 

• 2cos-1 x = cos-1 (2×2 – 1)
• 2sin-1x = sin-1 2x√(1 – x2)
• 3sin-1x = sin-1(3x – 4×3)
• 3cos-1 x = cos-1 (4×3 – 3x)
• 3tan-1x = tan-1((3x – x3/1 – 3×2))
• sin-1x + sin-1y = sin-1{ x√(1 – y2) + y√(1 – x2)}
• sin-1x – sin-1y = sin-1{ x√(1 – y2) – y√(1 – x2)}
• cos-1 x + cos-1 y = cos-1 [xy – √{(1 – x2)(1 – y2)}]
• cos-1 x – cos-1 y = cos-1 [xy + √{(1 – x2)(1 – y2)}
• tan-1 x + tan-1 y = tan-1(x + y/1 – xy)
• tan-1 x – tan-1 y = tan-1(x – y/1 + xy)
• tan-1 x + tan-1 y +tan-1 z = tan-1 (x + y + z – xyz)/(1 – xy – yz – zx)

Inverse trigonometric functions range and domain table

Inverse trigonometric functions problems and solutions 

Problem 1- What is the value of sin-1 (sin (4))? 

Solution 1- As we know, sin-1 (sin x) = x 

Therefore, the value of sin-1 (sin (4)) = 4 

Problem 2- Prove that tan-1211 + tan-1724= tan-112 

Solution 2- Tan-1x + Tan-1y = Tan-1x+y1-xy 

tan-1211 + tan-1724= tan-1211+7241-211724 

= tan-1 48+7724×1111×24-1424×11 = tan-1 125250 

= tan-112 

Therefore, we can verify that tan-1211 + tan-1724= tan-112 

Problem 3 – What is the principal value of sin-1-12? 

Solution 3 –  

We know that for all values of x belonging in a range of -1 to 1, Sin-1(-x) = – sin-1 x.

Therefore, let y = sin-1-12

Since, sin 6 = 12 

Therefore, sin-112 = 6

So, y = sin-1-sin 6 = 6 

Therefore, the principal value of sin-1-12 = 6

Conclusion

The branch of mathematics that deals with the angles and sides is called trigonometry. 

The concept of inverse trigonometry deals with the inverse functions of the trigonometric functions. Therefore, the inverse trigonometric functions are inverse cotangent, inverse cosecant, inverse sine, inverse tangent, inverse secant, and inverse cosine. 

When only two sides of a right triangle are known, the inverse trigonometric functions determine the angle measure.  

The concept of inverse trigonometric functions is generally used in physics, geometry, engineering, etc. Inverse trigonometric functions are also known as anti-trigonometric functions or arcus functions.