# Properties of inverse trigonometric functions I

Property I:

• sin¯¹ (sin θ) = θ                    for all θ ϵ [-π / 2, π / 2]
• cos¯¹ (cos θ) = θ                   for all θ ϵ [0, π]
• tan¯¹ (tan θ) = θ                   for all θ ϵ [-π / 2, π / 2]
• cosec¯¹ (cosec θ) = θ           for all θ ϵ [-π / 2, π / 2]
• sec¯¹ (sec θ) = θ                   for all θ ϵ [0, π], θ ≠ π / 2
• cot¯¹ (cot θ) = θ                   for all θ ϵ [0, π]

Property II:

• sin (sin¯¹ x) = x                 for all x ϵ [-1, 1]
• cos (cos¯¹ x) = x                for all x ϵ [-1, 1]
• tan (tan¯¹ x) = x                for all x ϵ R
• cosec (cosec¯¹ x) = x        for all x ϵ [-∞, -1] U [1, ∞]
• sec (sec¯¹ x) = x                 for all x ϵ [-∞, -1] U [1, ∞]
• cot (cot¯¹ x) = x                 for all x ϵ R

REMARK:  It should be noted that sin¯¹ (sin θ) ≠ θ, if all θ ∉ [-π / 2, π / 2]. In fact, we have.

{{\sin }^{-1}}\left( \sin \theta \right)=\left\{ \begin{align}& -\pi -\theta ,if\,\theta \in \left[ -3\pi /2,-\pi /2 \right] \\& \theta ,if\,\theta \in \left[ -\pi /2,\pi /2 \right] \\& \pi -\theta ,if\,\theta \in \left[ \pi /2,\,3\pi /2 \right] \\& -2\pi +\theta ,if\,\theta \in \left[ 3\pi /2,5\pi /2 \right] \\\end{align} \right. and so on.

Similarly, we have

{{\cos }^{-1}}\left( \cos \theta \right)=\left\{ \begin{align}& -\theta ,if\,\theta \in \left[ -\pi ,0 \right] \\& \theta ,if\,\theta \in \left[ 0,\pi \right] \\& 2\pi -\theta ,if\,\theta \in \left[ \pi ,2\pi \right] \\& -2\pi +\theta ,if\,\theta \in \left[ 2\pi ,3\pi \right] \\\end{align} \right. and so on.

{{\tan }^{-1}}\left( \tan \theta \right)=\left\{ \begin{align}& \pi -\theta ,if\,\theta \in \left[ -3\pi /2,-\pi /2 \right] \\& \theta ,if\,\theta \in \left[ -\pi /2,\pi /2 \right] \\& \theta -\pi ,if\,\theta \in \left[ \pi /2,3\pi /2 \right] \\& \theta -2\pi ,if\,\theta \in \left[ 3\pi /2,5\pi /2 \right] \\\end{align} \right. and so on.

Property III:

•  sin¯¹ (-x) = – sin¯¹ x,                     for all x ϵ [-1, 1]
• cos¯¹ (-x) = π – cos¯¹ x,                  for all x ϵ [-1, 1]
• tan¯¹ (-x) = – tan¯¹ x,                     for all x ϵ R
• cosec¯¹ (-x) = – cosec¯¹ x,             for all x ϵ [-∞, -1] U [1, ∞]
• sec¯¹ (-x) = π – sec¯¹ x,                  for all x ϵ [-∞, -1] U [1, ∞]
• cot¹ (-x) = π – cot¯¹ x,                    for all x ϵ R.

Property IV:

(i) sin¯¹ (1/x) = cosec¯¹ x,                     for all x ϵ [-∞, -1] U [1, ∞]

(ii) cos¯¹ (1/x) = sec¯¹ x,                       for all x ϵ [-∞, -1] U [1, ∞]

(iii) {{\tan }^{-1}}\left( \frac{1}{x} \right)=\left\{ \begin{align}& {{\cot }^{-1}}xfor\,x>0 \\& -\pi +{{\cot }^{-1}}xfor\,x<0 \\\end{align} \right.

Property V:

• sin-¹ x + cos¯¹ x = π/2,                     for all x ϵ [-1, 1]
• tan¯¹ x + cot¯¹ x = π/2,                   for all x ϵ R
• sec ¹ x + cosec x = π/2,                for all x ϵ [-∞, -1] U [1, ∞]

Property VI:

(i) {{\tan }^{-1}}x+{{\tan }^{-1}}y=\left\{ \begin{align}& {{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right),if\,xy<1 \\& \pi +{{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right),if\,x>0,y>0\,and\,xy>1 \\& -\pi +{{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right),if\,x<0,y>0\,and\,xy>1 \\\end{align} \right..

(ii) {{\tan }^{-1}}x-{{\tan }^{-1}}y=\left\{ \begin{align}& {{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right),if\,xy>-1 \\& \pi +{{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right),if\,x>0,y>0\,and\,xy<-1 \\& =\pi +{{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right),if\,x<0,y>0\,and\,xy<-1 \\\end{align} \right..

Property VII:

(i) {{\sin }^{-1}}x+{{\sin }^{-1}}y=\left\{ \begin{align}& {{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right),if\,-1\le x,y\le 1\,and\,{{x}^{2}}+{{y}^{2}}\le 1 \\& or \\& \,\,\,\,\,\,\,\,if\,xy<0\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ & \pi -{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right),if\,0<x,y\le 1\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ & -p-{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right),if\,-1\le x,y<0\,and\,{{x}^{2}}+{{y}^{2}}>1 \\\end{align} \right..

(ii) {{\sin }^{-1}}x-{{\sin }^{-1}}y=\left\{ \begin{align}& {{\sin }^{-1}}\,\left( x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}} \right),if\,-1\le \,x,y\le \,1\,and\,{{x}^{2}}+{{y}^{2}}\le 1 \\& or \\& if\,xy>0\,and\,{{x}^{2}}+{{y}^{2}}>\,1 \\& \pi -{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}} \right),if\,0<x\le 1,-1\le y\le 0\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ & -\pi -{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}} \right)-y\sqrt{1-{{x}^{2}}},if\,-1\le x<0,0<y\le 1\,and\,{{x}^{2}}+{{y}^{2}}>1 \\\end{align} \right..