Inverse trigonometric ratios of Multiple Angles

Inverse trigonometric ratios of Multiple Angles

Komal MiglaniUpdated on 02 Jul 2025, 07:30 PM IST

Prior to understanding the multiple angles of inverse trigonometric function, let's discuss inverse trigonometric function. Inverse trigonometric functions can be defined as the inverses of the basic trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant. The multiple-angle formulas for these trigonometric functions can also be represented in the form of their inverse functions.

This Story also Contains

  1. What are Multiple angles in terms of inverse trigonometric function?
  2. Multiple angles in terms of arcsin
  3. Multiple angles in terms of arccos
  4. Multiple angles in terms of arctan and arcsin
  5. Multiple angles in terms of arctan and arccos
Inverse trigonometric ratios of Multiple Angles
Inverse trigonometric ratios of Multiple Angles

In this article, we will cover the concept of Multiple angles. This category falls under the broader category of Trigonometry, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of six questions have been asked on this topic including one in 2014, one in 2019, two in 2021, and two in 2023.

What are Multiple angles in terms of inverse trigonometric function?

The multiple angle formula is the representation of the angle which is multiple of a given angle. We can calculate the values of multiple angles by expressing each trigonometric function in its expanded form.

Multiple angles in terms of arcsin

1. Double of Inverse Trigonometric Function Formulas

$2 \sin ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), & \frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} \\ \pi-\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), & x>\frac{1}{\sqrt{2}} \\ -\pi-\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), & x<-\frac{1}{\sqrt{2}}\end{array}\right.$
2. Triple of Inverse Trigonometric Function Formulas

$3 \sin ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\sin ^{-1}\left(3 x-4 x^3\right), & -\frac{1}{2} \leq x \leq \frac{1}{2} \\ \pi-\sin ^{-1}\left(3 x-4 x^3\right), & x>\frac{1}{2} \\ -\pi-\sin ^{-1}\left(3 x-4 x^3\right) & x:-\frac{1}{2}\end{array}\right.$

Multiple angles in terms of arccos

1. Double of Inverse Trigonometric Function Formulas

$2 \cos ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\cos ^{-1}\left(2 x^2-1\right), & \text { if } 0 \leq x \leq 1 \\ 2 \pi-\cos ^{-1}\left(2 x^2-1\right), & \text { if }-1 \leq x \leq 0\end{array}\right.$
2. Triple of Inverse Trigonometric Function Formulas

$3 \cos ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\cos ^{-1}\left(4 x^3-3 x\right), & \text { if } \frac{1}{2} \leq x \leq 1 \\ 2 \pi-\cos ^{-1}\left(4 x^3-3 x\right), & \text { if }-\frac{1}{2} \leq x \leq \frac{1}{2} \\ 2 \pi+\cos ^{-1}\left(4 x^3-3 x\right), & \text { if }-1 \leq x \leq-\frac{1}{1}\end{array}\right.$

Multiple angles in terms of arctan and arcsin

The multiple-angle formula of arctan in terms of arcsin is given by

$2 \tan ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\sin ^{-1}\left(\frac{2 x}{1+x^2}\right), & \text { if }-1 \leq x \leq 1 \\ \pi-\sin ^{-1}\left(\frac{2 x}{1+x^2}\right), & \text { if } x>1 \\ -\pi-\sin ^{-1}\left(\frac{2 x}{1+x^2}\right), & \text { if } x<-1\end{array}\right.$

Multiple angles in terms of arctan and arccos

The multiple angle formula of arctan in terms of arccos is given by

$2 \tan ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), & \text { if } 0 \leq x<\infty \\ -\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), & \text { if }-\infty<x \leq 0\end{array}\right.$

Recommended Video Based on Inverse Trigonometric of Multiple Angles :

Solved Examples Based on Multiple Angles in Terms of Inverse Trigonometric Functions

Example 1: $S=\left\{x \in R ; 0<x<1\right.$ and $\left.2 \tan ^{-1}\left(\frac{1-x}{1+x}\right)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$ If n(S) denotes the number of elements in S then : [JEE MAINS 2023]

Solution: Put $x=\tan \theta \quad \theta \in\left(0, \frac{\pi}{4}\right)$

$\begin{aligned} & 2 \tan ^{-1}\left(\frac{1-\tan \theta}{1+\tan \theta}\right)=\cos ^{-1}\left(\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\right) \\ & 2 \tan ^{-1}\left[\tan \left(\frac{\pi}{4}-\theta\right)\right]=\cos ^{-1}[\cos (2 \theta)] \\ & \Rightarrow 2\left(\frac{\pi}{4}-\theta\right)=2 \theta \Rightarrow \theta=\frac{\pi}{8} \\ & \Rightarrow x=\tan \frac{\pi}{8}=\sqrt{2}-1 \simeq 0.414\end{aligned}$

Hence, the answer is $n(S)=2$ and only one element in S is less then $\frac{1}{2}$

Example 2: If the sum of all the solutions of $\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\frac{\pi}{3},-1<x<1, x \neq 0{ }_{\text {is }} \alpha-\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to
[JEE MAINS 2023]

Solution$x \in(-1,1) \quad \tan ^{-1}\left(\frac{2 \mathrm{x}}{1-\mathrm{x}^2}\right)=2 \tan ^{-1} \mathrm{x}$
$x \in(0,1) \quad \cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)=2 \tan ^{-1} x$
$x \in(-1,0) \quad \cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\mathrm{IT}+\tan ^{-1}\left(\frac{2 x}{1-\mathrm{x}^2}\right)=\mathrm{IT}+2 \tan ^{-1} \mathrm{x}$$x \in(0,1) \quad 2 \tan ^{-1} x+2 \tan ^{-1} x=\pi / 3$

$\tan ^{-1} x=\frac{\pi}{12}$

$\mathrm{x}=2-\sqrt{3}$$\begin{gathered}x \in(-1,0) \quad 2 \tan ^{-1} x+I T+2 \tan ^{-1} x=\pi / 3 \\ 4 \tan ^{-1} x=\frac{-2 \pi}{3} \\ \tan ^{-1} \mathrm{x}=-\pi / 6 \\ \mathrm{x}=-1 / \sqrt{3} \\ (2-\sqrt{3})+\left(-\frac{1}{\sqrt{3}}\right)=a-\frac{4}{\sqrt{3}} \\ 2-4 / \sqrt{3}=\alpha-4 / \sqrt{3} \\ \alpha=2\end{gathered}$
Hence, the answer is 2.

Example 3: $\operatorname{cosec}\left[2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right]$ is equal to: [JEE MAINS 2021]

Solution
$
\begin{aligned}
& \operatorname{cosec}\left[2 \tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{3}{4}\right)\right] \\
& \operatorname{cosec}\left[\tan ^{-1}\left(\frac{5}{12}\right)+\tan ^{-1}\left(\frac{3}{4}\right)\right] \\
& \operatorname{cosec}\left[\tan ^{-1}\left(\frac{\frac{5}{12}+\frac{3}{4}}{1-\frac{5}{12} \times \frac{3}{4}}\right)\right] \\
& =\operatorname{cosec}\left[\tan ^{-1}\left(\frac{56}{33}\right)\right]=\frac{65}{56}
\end{aligned}
$

$
\text { Hence, the answer is } \frac{65}{56}
$

Example 4: The sum of possible values of x for $\tan ^{-1}(x+1)+\cot ^{-1}\left(\frac{1}{x-1}\right)=\tan ^{-1}\left(\frac{8}{31}\right)$ is : [JEE MAINS 2021]

Solution
$
\tan ^{-1}(x+1)+\cot ^{-1}\left(\frac{1}{x-1}\right)=\tan ^{-1} \frac{8}{31}
$

Taking tangent both sides :-

$
\begin{aligned}
& \frac{(x+1)+(x-1)}{1-\left(x^2-1\right)}=\frac{8}{31} \\
& \Rightarrow \frac{2 x}{2-x^2}=\frac{8}{31} \\
& \Rightarrow 4 x^2+31 x-8=0 \\
& \Rightarrow x=-8, \frac{1}{4}
\end{aligned}
$

$\begin{aligned} & \text { But, if } x=\frac{1}{4} \\ & \tan ^{-1}(x+1) \in\left(0, \frac{\pi}{2}\right) \\ & \& \cot ^{-1}\left(\frac{1}{x-1}\right) \in\left(\frac{\pi}{2}, \pi\right)\end{aligned}$
$
\Rightarrow L H S>\frac{\pi}{2} \quad \& \quad R H S<\frac{\pi}{2}
$

(Not possible)
Hence, $x=-8$

Hence, the answer is $-\frac{32}{4}$

Example 5: If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^2, x \in\left(0, \frac{\pi}{2}\right)$ $\square$ then $\overline{d x}$ is equal to [JEE MAINS 2019]

Solution: Important Results of Inverse Trigonometric Functions $\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}$ when $x \in R$

$2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^2 \quad x \in\left(0, \frac{\pi}{2}\right)$

$2 y=\left(\cot ^{-1}\left(\frac{\frac{\sqrt{3} \cos x}{\cos x}+\frac{\sin x}{\cos x}}{\frac{\cos x}{\cos x}-\frac{\sqrt{3} \sin x}{\cos x}}\right)\right)^2$

$=\left(\cot ^{-1}\left(\frac{\sqrt{3}+\tan x}{1-\sqrt{3} \tan x}\right)\right)^2$

$=\left(\cot ^{-1}\left(\tan \left(\frac{\pi}{3}+x\right)\right)\right)^2$

$=\left(\frac{\pi}{2}-\tan ^{-1}\left(\tan \left(\frac{\pi}{3}+x\right)\right)\right)^2$

$=\left(\frac{\pi}{2}-\left(\frac{\pi}{3}+x\right)\right)^2$

$=\left(\frac{\pi}{2}-\frac{\pi}{3}-x\right)^2=\left(\frac{\pi}{6}-x\right)^2$

$2 y=\left(x-\frac{\pi}{6}\right)^2$

$\frac{d y}{d x}=\frac{1}{2} \times 2\left(x-\frac{\pi}{6}\right)$

$=x-\frac{\pi}{6}$

Hence, the answer is $x-\frac{\pi}{6}$


Frequently Asked Questions (FAQs)

Q: How does the slope of arcsin(3x) compare to the slope of arcsin(x) at x = 0?
A:
The slope of arcsin(3x) at x = 0 is three times the slope of arcsin(x) at x = 0. This is because the derivative of arcsin(3x) is 3/√(1-(3x)^2), which equals 3 at x = 0, while the derivative of arcsin(x) is 1/√(1-x^2), which equals 1 at x = 0.
Q: What is the double angle formula for arcsec?
A:
The double angle formula for arcsec is:
Q: How does the concept of inverse trigonometric ratios of multiple angles relate to complex numbers?
A:
Inverse trigonometric ratios of multiple angles can be extended to complex numbers, where they exhibit interesting properties and relationships. These complex extensions are useful in various areas of mathematics and physics, including the study of conformal mappings.
Q: What is the relationship between arccot(x/2) and arccot(x)?
A:
For x > 0, we can express arccot(x/2) in terms of arccot(x) as:
Q: How does the derivative of arcsin(2x) relate to the derivative of arcsin(x)?
A:
The derivative of arcsin(x) is 1/√(1-x^2), while the derivative of arcsin(2x) is 2/√(1-(2x)^2) = 1/√(1-4x^2). The factor of 2 in arcsin(2x) results in a more rapidly changing derivative as x approaches ±1/2.
Q: What is the relationship between arccos(x/2) and arccos(x)?
A:
For -1 ≤ x ≤ 1, we can express arccos(x/2) in terms of arccos(x) as:
Q: What is the domain of arctan(3x) and how does it compare to arctan(x)?
A:
The domain of arctan(x) is all real numbers, and the domain of arctan(3x) is also all real numbers. Unlike arcsin and arccos, the domain of arctan is not restricted when considering multiple angles.
Q: How does the concept of inverse trigonometric ratios of multiple angles relate to power series expansions?
A:
Inverse trigonometric ratios of multiple angles can be expressed as power series. These series are often more complex than those for single angles and can provide insights into the behavior of these functions for small values of x.
Q: How does the inverse tangent of half-angles relate to the inverse tangent of the original angle?
A:
For -1 < x < 1, we can express arctan(x/2) in terms of arctan(x) as:
Q: What is the significance of the factor 3 in arccos(3x)?
A:
The factor 3 in arccos(3x) effectively "compresses" the input horizontally by a factor of 3. This means that the function reaches its extreme values (0 and π) more quickly as x increases or decreases, compared to arccos(x).