Inverse Trigonometric Functions

Inverse Trigonometric Functions, also known as arcus, anti-trigonometric, or cyclometric functions, are the inverse counterparts of the fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. They are instrumental in determining angles based on trigonometric ratios. Widely utilized in engineering, Physics, Geometry, and navigation, these functions play pivotal roles in various applications.

Inverse trigonometry functions are crucial in Mathematics, especially in Calculus and Trigonometry. They are the inverse operations of trigonometric functions and have diverse applications in solving equations, finding angles, and analyzing periodic phenomena. In this thorough exploration, we will delve deep into inverse trigonometric functions, exploring their formulas, derivatives, graphs, properties, and much more.

1.0What Are Inverse Trigonometric Functions?

Inverse trigonometric functions are functions that yield angles when given the ratios of sides in a right triangle. They are denoted by sin⁻¹x, cos⁻¹x, tan⁻¹x, cot⁻¹x, sec⁻¹x, and cosec⁻¹x. These functions have restricted domains and ranges to ensure their inverse nature.

2.0Graphs of Inverse Trigonometric Functions:

The graphs of inverse trigonometric functions exhibit unique characteristics due to their restricted domains and ranges. Here are the graphs of the six primary inverse trigonometric functions:

1. Graph of Arcsine Function (sin⁻¹x):

2. Graph of Arccosine Function (cos⁻¹x):

3. Graph of Arctangent Function (tan⁻¹x):

4. Graph of Arc cotangent Function (cot⁻¹x):

5. Graph of Arcsecant Function (sec⁻¹x):

6. Graph of Arccosecant Function (cosec⁻¹x):

3.0Domain and Range of Inverse Trigonometric Functions:

The domain and range of inverse trigonometric functions are determined by their definitions and properties. For example, the domain of sin^–1x is [–1,1], and its range is $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$. Understanding these domains and ranges is essential for solving equations and analyzing functions.

Here are the domain and range of the six primary inverse trigonometric functions:

Domain of Inverse Trigonometric Functions

The domain of inverse trigonometric functions is limited to ensure their inverse nature. For example, the domain of sin⁻¹x and cos⁻¹x is [−1, 1], while the domain of tan⁻¹x is R. Understanding these domains is crucial for determining where the functions are defined. Domains of the six primary inverse trigonometric functions are given above.

Range of Inverse Trigonometric Functions

The range of inverse trigonometric functions varies depending on the function. For example, the range of sin^–1 x is $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$, while the range of cos^–1 x is [0, π] These ranges reflect the restrictions imposed on the outputs of inverse trigonometric functions. Range of the six primary inverse trigonometric functions are given above.

4.0Inverse Trigonometric Functions Identities

Inverse trigonometric functions have several identities that relate them to their corresponding trigonometric functions. For example, sin(sin⁻¹x) = x and sin⁻¹ (sin x) = x in specific domains are identities that demonstrate the relationship between sine and arcsine. These identities are useful in simplifying expressions and solving equations.

5.0Properties of Inverse Trigonometric Functions

Inverse trigonometric functions possess various properties that govern their behavior. These properties include symmetry, periodicity, and monotonicity. For instance, sin⁻¹(−x) = −sin⁻¹x reflects the odd symmetry of arcsine function. Understanding these properties enhances our comprehension of inverse trigonometric functions.

Property – 1

(i) ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$; |x| < 1

(ii) ${\tan }^{-1}\mathrm{x}+{\cot }^{-1}\mathrm{x}=\frac{\pi }{2}$ ; x ∈ R

(iii) ${\mathrm{cosec}}^{-1}x+{\sec }^{-1}x=\frac{\pi }{2}$; |x| ≥ 1

Property – 2

(i) sin^–1(–x) = –sin ^–1x;               |x| ≤ 1

(ii) cosec^–1(–x) = – cosec^–1x |x| > 1

(iii) tan^–1 (–x) = –tan^–1x;                 x ∈ R

(iv) cot^–1 (–x) = π – cot^–1 x;             x∈ R 

(v) cos^–1 (–x) = π – cos^–1x;         |x| < 1

(vi) sec^–1 (–x) = π – sec^–1 x;   |x| > 1

Property – 3

(i) ${\mathrm{cosec}}^{-1}\mathrm{x}={\sin }^{-1}\frac{1}{\mathrm{x}}$; |x| ≥1

(ii) ${\sec }^{-1}x={\cos }^{-1}\frac{1}{x}$; |x| ≥ 1

(iii) ${\cot }^{-1}x=\{\begin{array}{lll}{\tan }^{-1}\frac{1}{x}& ;& x>0\\ \pi +{\tan }^{-1}\frac{1}{x}& ;& x<0\end{array}$

Property – 4

(i) (a) ${\tan }^{-1}x+{\tan }^{-1}y=\{\begin{array}{ccc}{\tan }^{-1}\frac{x+y}{1-xy}& \text{ where }& x>0,y>0\&xy<1\\ \pi +{\tan }^{-1}\frac{x+y}{1-xy}& \text{ where }& x>0,y>0\&xy>1\\ \frac{\pi }{2},& \text{ where }& x>0,y>0\&xy=1\end{array}$

  (b) ${\tan }^{-1}\mathrm{x}-{\tan }^{-1}\mathrm{y}={\tan }^{-1}\frac{x-y}{1+xy}wherex>0,y>0$

(c) ${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z={\tan }^{-1}\left[\frac{x+y+z-xyz}{1-xy-yz-zx}\right]$

if x > 0, y > 0, z > 0 and xy + yz + zx < 1

(ii) (a) ${\sin }^{-1}x+{\sin }^{-1}y=$

$\{\begin{array}{ll}{\sin }^{-1}\left[x\sqrt{1-y^2}+y\sqrt{1-x^2}\right]& \text{ where }x>0,y>0\&\left(x^2+y^2\right)\le 1\\ \pi -{\sin }^{-1}\left[x\sqrt{1-y^2}+y\sqrt{1-x^2}\right]& \text{ where }x>0,y>0\&x^2+y^2>1\end{array}$

(b) ${\sin }^{-1}-{\sin }^{-1}y={\sin }^{-1}\left[x\sqrt{1-y^2}-y\sqrt{1-x^2}\right]wherex>0,y>0$

(iii) (a) ${\cos }^{-1}x+{\cos }^{-1}y={\cos }^{-1}\left[xy-\sqrt{1-x^2}\sqrt{1-y^2}\right]wherex>0,y>0$

(b) ${\cos }^{-1}\mathrm{x}-{\cos }^{-1}\mathrm{y}=\{\begin{array}{lc}{\cos }^{-1}\left(xy+\sqrt{1-x^2}\sqrt{1-y^2}\right);& x<y,x,y>0\\ -{\cos }^{-1}\left(xy+\sqrt{1-x^2}\sqrt{1-y^2}\right);& x>y,x,y>0\end{array}$

6.0Inverse Trigonometric Functions Solved Problems

Example 1: The value of ${\sin }^{-1}(-\sqrt{3}/2)$ is -

(A) –π/3 (B) –2π/3 (C) 4π/3 (D) 5π/3

Ans. (A)

Solution: 

${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=-{\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=-\frac{\pi }{3}$

Example 2: Domain of the function $y=\sqrt{{\sin }^{-1}\left({\log }_{2}x\right)}$

(A) [1, 2] (B) [–1, 2] (C) [–2, 2] (D) [–3, 3]

Ans. (A)

Solution: sin^–1(log₂ x) ≥ 0 and –1 ≤ log₂ x ≤ 1 and  x > 0

$0\le {\sin }^{-1}\left({\log }_{2}x\right)\le \frac{\pi }{2}$

0 ≤ (log₂ x) ≤ 1

1 ≤ x ≤ 2

D = [1, 2]

Example 3: Range of sin^–1 x + cosec^–1 x is -

(A) {–π, π} (B) (-π, π)

(C) $\left\{0,\frac{\pi }{2}\right\}$ (D) none of these

Ans. (A)

Solution: Domain of sin^–1 x is [–1, 1]

domain of cosec^–1 x is (–∞, –1] ∪ [1, ∞)

The common domain is {–1, 1} only.

range = sin^–1 (–1) + cosec^–1 (–1) = –π

or sin^–1 (1) + cosec^–1 (1) =π

Hence, range = {–π, π}

Example 4: Find domain and range of y = sin^–1 (x) + cos^–1 (x) + tan^–1 (x) 

Solution: D_f : x ∈ [–1,1]  and x ∈ [–1,1]  and x ∈ R

⇒ x ∈ [–1,1]

∴ D_f : [–1,1] 

$\mathrm{f}(\mathrm{x})=\frac{\pi }{2}+{\tan }^{-1}(x)$

–1 ≤ x ≤ –1

$\Rightarrow -\frac{\pi }{4}\le {\tan }^{-1}x\le \frac{\pi }{4}$

$\Rightarrow \frac{\pi }{4}\le \frac{\pi }{2}+{\tan }^{-1}x\le \frac{3\pi }{4}$

$\therefore R_f:\left[\frac{\pi }{4},\frac{3\pi }{4}\right]$

Example 5: Prove that sec²(tan^–1 2) + cosec² (cot^–1) 3) = 15

Solution: We have,

sec²(tan^-12)   + cosec² (cot^–1 3)

$={\left\{\sec \left({\tan }^{-1}2\right)\right\}}^2+{\left\{\mathrm{cosec}\left({\cot }^{-1}3\right)\right\}}^2$

$={\left\{\sec \left({\tan }^{-1}\frac{2}{1}\right)\right\}}^2+{\left\{\mathrm{cosec}\left({\cot }^{-1}\frac{3}{1}\right)\right\}}^2$

$={\left\{\sec \left({\sec }^{-1}\sqrt{5}\right)\right\}}^2+\{cosec{\left({cosec}^{-1}\sqrt{10}\right\}}^2$

$={\left\{\sec \left({\sec }^{-1}\sqrt{5}\right)\right\}}^2+\{\mathrm{cosec}{\left({\mathrm{cosec}}^{-1}\sqrt{10}\right\}}^2$

$=(\sqrt{5}{)}^2+(\sqrt{10}{)}^2=15$

7.0Sample Questions on Inverse Trigonometric Functions

Q. What are inverse trigonometric functions?

Ans: Inverse trigonometric functions are functions that provide angles given the ratios of sides in a right triangle. They are denoted by sin⁻¹x, cos⁻¹x, tan⁻¹x, cot⁻¹x, sec⁻¹x, and cosec⁻¹x.

Q. What are some common identities involving inverse trigonometric functions?

Ans: Common identities include sin(sin⁻¹x) = x, cos(cos⁻¹x) = x, and tan(tan⁻¹x) = x. These identities express the relationships between trigonometric functions and their inverses.