Trigonometry-Inverse Trigonometric Functions

The inverse trigonometric functions are commonly described as the inverse functions of the trigonometric functions. In other words, they are the inverse of the essential trigonometric functions such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec) functions. 

 As we know, if f(x) is a bijective function s.t.

 f(x) : [a,b] → [c,d]

 f(x) = y, x ∈ [a,b] and y ∈ [c,d]

 Then, we can always define a function in opposite directions s.t. 

 f-1(x) : [c,d] → [a,b] 

 f-1(y) : x, x ∈ [a,b] and y ∈ [c,d] 

 Hence, f-1(x) is called the inverse of the function. 

Domain and Range of Inverse Trigonometric Functions 

 The domain and range of trigonometric functions donate the input and output values of trigonometric functions. In simple words, the range of the original function is the domain of the inverse function. 

 1. If sin y = x, then y = sin-1 x

 Under certain conditions, -1 ≤ sin y ≤ 1; sin y = x.     ∴ -1 ≤ x ≤ 1

 Thus, sin y = -1 → y = -π/2 and sin y = 1 → y = π/2 

Hence, numerically smallest angles or real numbers. ∴ -π/2 ≤ y ≤ π/2

 The restrictions on the values of x and y deliver us with the domain and range for the 

function y = sin-1 x.  

i.e., Domain: x ∈ [-1,1]

Range: y ∈ [-π/2,π/2]

 2. If cos y = x, then y = cos-1 x

 Under certain conditions, -1 ≤ cos y ≤ 1 → -1 ≤ x ≤ 1

 Thus, cos y = -1 → y = π and cos y = 1 → y = 0

 ∴ 0 ≤ y ≤ π (as cos x is a decreasing function in [0,π]; 

Hence, cos π ≤ cos y ≤ cos 0 

The restrictions on the values of x and y deliver us with the domain and range for the function y = cos-1 x. 

 i.e., Domain: x ∈ [-1,1] 

Range: y ∈ [0,π]

 3. If tan y = x, then y = tan-1 x

 Here, tan y ∈ R → x ∈ R;

– ∞ ≤ tan y ≤ ∞ → -π/2 ≤ y ≤ π/2

 i.e., Domain: x ∈ R

Range: y ∈ [-π/2,π/2]

 4. If cot y = x, then y = cot-1 x

 Here, cot y ∈ R → x ∈ R;

– ∞ ≤ cot y ≤ ∞ → 0 ≤ y ≤ π

 i.e., Domain: x ∈ R

Range: y ∈ [0,π]

 5. If sec y = x, then y = sec-1 x 

 Where x ≥ 1 and 0 ≤ y ≤ π; y ≠ π/2

 i.e., Domain: x ∈ R -[-1,1]

Range: y ∈ [0,π] – [π/2]

 6. If cosec y = x, then y = cosec-1 x

 Where x ≥ 1 and -π/2 ≤ y ≤ π/2; y ≠ 0

 i.e., Domain: ∈ R -[-1,1] 

Range: ∈ [-π/2,π/2] – [0]

Table of Domain and Range of Inverse Trigonometric Functions

 The below-given table discusses the domain and range of all the six inverse trigonometric functions:

Sum and Difference of Inverse Trigonometric Functions:

 The sum and difference of inverse trigonometric functions involve the sum and difference of two different angles. For instance, below are the sum and difference of angles in terms of tan-1(x), sin-1(x) and cos-1(x). 

Sum and Difference of Angles in Inverse Tangent [Tan^(-1)]

• 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 (2 x)/(1 – x2) = 2 tan-1 x

 Sum and Difference of Angles in Inverse Sine [Sin-1]

• sin-1 x + sin-1 y = sin-1 [x √(1 – y2) + y √(1 – x2)]

• sin-1 x – sin-1 y = sin-1 [x √(1 – y2) – y √(y – x2)]

• sin-1 (2 x)/(1 + x2) = 2 tan-1 x

 Sum and Difference of Angles in Inverse Cosine [Cos-1]

• 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)] 

• cos-1 (1 – x2)/(1 + x2) = 2 tan-1 x

Conclusion

 The primary purpose of inverse trigonometric functions is to obtain an angle from any trigonometric ratio. We have explained all the essential formulas of inverse trigonometric functions in this topic. To get a detailed understanding of the concept, read some other similar topics of inverse trigonometric functions like derivatives, integrals, complementary angles, and multiple angles.