Trig Functions - All Best Math | Trigonometric Funtions 500+

Hello, friends, today we will learn about Trig Functions. And with this, we will study thoroughly about other topics of trigomometrics like Inverse Trigonometric Functions, Calculus (Differentiation & Integration), Subtituions, Definite Integral, Integration of Algebraic Functions etc.

Table of Contents

Trigonometric Functions

So first we start with the basic formula for trigonometric functions.

Basic Trigonometric Functions

$Sin2A=2SinACosA$

$Sin2A=\frac { 2TanA }{ 1+{ Tan }^{ 2 }A }$

$Cos2A=1-2{ Sin }^{ 2 }A$

$Cos2A=2{ Cos }^{ 2 }A-1$

$Cos2A={ Cos }^{ 2 }A-{ Sin }^{ 2 }A$

$Cos2A=\frac { 1-{ Tan }^{ 2 }A }{ 1+{ Tan }^{ 2 }A }$

$Tan2A=\frac { 2TanA }{ 1-{ Tan }^{ 2 }A}$

$Sin3A=3SinA-4{ Sin }^{ 3 }A$

$Cos3A=4{ Cos }^{ 3 }A-3CosA$

$Tan3A=\frac { 3TanA-{ Tan }^{ 3 }A }{ 1-3{ Tan }^{ 2 }A}$

$Sin(A+B)Sin(A-B)={ Sin }^{ 2 }A-{ Sin }^{ 2 }B$

$Sin(A+B)Sin(A-B)={ Cos }^{ 2 }B-Cos^{ 2 }A$

$Cos(A+B)Cos(A-B)={ Cos }^{ 2 }A-Sin^{ 2 }B$

$Cos(A+B)Cos(A-B)={ Cos }^{ 2 }B-Sin^{ 2 }A$

$SinA=2Sin\frac { A }{ 2 } Cos\frac { A }{ 2 }$

$SinA=\frac { 2Tan\frac { A }{ 2 } }{ 1+{ Tan }^{ 2 }\frac { A }{ 2 } }$

$CosA=1-2{ Sin }^{ 2 }\frac { A }{ 2 }$

$CosA=2{ Cos }^{ 2 }\frac { A }{ 2 } -1$

$CosA={ Cos }^{ 2 }\frac { A }{ 2 } -{ Sin }^{ 2 }\frac { A }{ 2 }$

$CosA=\frac { 1-{ Tan }^{ 2 }\frac { A }{ 2 } }{ 1+{ Tan }^{ 2 }\frac { A }{ 2 } }$

$TanA=\frac { 2{ Tan }\frac { A }{ 2 } }{ 1-{ Tan }^{ 2 }\frac { A }{ 2 } }$

$1+Cos2A=2{ Cos }^{ 2 }A$
$1-Cos2A=2{ Sin }^{ 2 }A$
$1+CosA=2{ Cos }^{ 2 }\frac { A }{ 2 }$
$1-CosA=2{ Sin }^{ 2 }\frac { A }{ 2 }$

Most Useable Trigonometric Functions

$Sin(A+B)=SinACosB+CosASinB$

$Sin(A-B)=SinACosB-CosASinB$

$Cos(A+B)=CosACosB-SinASinB$

$Cos(A-B)=CosACosB+SinASinB$

$Tan(A+B)=\frac { TanA+TanB }{ 1-TanATanB }$

$Tan(A-B)=\frac { TanA-TanB }{ 1+TanATanB }$

$Cot(A+B)=\frac { CotACotB-1 }{ CotB+CotA }$

$Cot(A-B)=\frac { CotACotB+1 }{ CotB-CotA }$

$2SinACosB=Sin(A+B)+Sin(A-B)$

$2CosASinB=Sin(A+B)-Sin(A-B)$

$2CosACosB=Cos(A+B)+Cos(A-B)$

$-2SinASinB=Cos(A+B)-Cos(A-B)$

$SinC+SinD=2Sin\frac { C+D }{ 2 } Cos\frac { C-D }{ 2 }$

$SinC-SinD=2Cos\frac { C+D }{ 2 } Sin\frac { C-D }{ 2 }$

$CosC+CosD=2Cos\frac { C+D }{ 2 } Cos\frac { C-D }{ 2 }$

$CosC-CosD=-2Sin\frac { C+D }{ 2 } Sin\frac { C-D }{ 2 }$

Inverse Trigonometric Functions

Definition :- If Sinθ = x then θ = Sin^-1x

$Sin^{ -1 }(Sin\theta )\quad =\quad \theta$

$Cos^{ -1 }(Cos\theta )\quad =\quad \theta$

$Tan^{ -1 }(Tan\theta )\quad =\quad \theta$

$Cot^{ -1 }(Cot\theta )\quad =\quad \theta$

$Sec^{ -1 }(Sec\theta )\quad =\quad \theta$

$Cosec^{ -1 }(Cosec\theta )\quad =\quad \theta$

$Sin(Sin^{ -1 }x)\quad =\quad x$

$Cos(Cos^{ -1 }x)\quad =\quad x$

$Tan(Tan^{ -1 }x)\quad =\quad x$

$Cot(Cot^{ -1 }x)\quad =\quad x$

$Sec(Sec^{ -1 }x)\quad =\quad x$

$Cosec(Cosec^{ -1 }x)\quad =\quad x$

$Sin\left( \frac { \pi }{ 2 } -\theta \right) \quad =\quad Cos\theta$

$Tan\left( \frac { \pi }{ 2 } -\theta \right) \quad =\quad Cot\theta$

$Sec\left( \frac { \pi }{ 2 } -\theta \right) \quad =\quad Cosec\theta$

$Sin^{ -1 }x+Cos^{ -1 }x=\frac { \pi }{ 2 }$

$Tan^{ -1 }x+Cot^{ -1 }x=\frac { \pi }{ 2 }$

$Sec^{ -1 }x+Cosec^{ -1 }x=\frac { \pi }{ 2 }$

Note :-

$Sin^{ -1 }x\quad \neq \quad \frac { 1 }{ Sinx }$
$(Sinx)^{ -1 }\quad =\quad \frac { 1 }{ Sinx }$
$Sinx^{ -1 }\quad =\quad Sin\frac { 1 }{ x }$

$Sin\theta \quad =\quad \frac { 1 }{ Cosec\theta }$

$Cos\theta \quad =\quad \frac { 1 }{ Sec\theta }$

$Tan\theta \quad =\quad \frac { 1 }{ Cot\theta }$

$Cot\theta \quad =\quad \frac { 1 }{ Tan\theta }$

$Sec\theta \quad =\quad \frac { 1 }{ Cos\theta }$

$Cosec\theta \quad =\quad \frac { 1 }{ Sin\theta }$

${ Sin }^{ -1 }x\quad =\quad { Cosec }^{ -1 }\frac { 1 }{ x }$

${ Cos }^{ -1 }x\quad =\quad { Sec }^{ -1 }\frac { 1 }{ x }$

${ Tan }^{ -1 }x\quad =\quad { Cot }^{ -1 }\frac { 1 }{ x }$

${ Cot }^{ -1 }x\quad =\quad Tan^{ -1 }\frac { 1 }{ x }$

${ Sec }^{ -1 }x\quad =\quad Cos^{ -1 }\frac { 1 }{ x }$

${ Cosec }^{ -1 }x\quad =\quad Sin^{ -1 }\frac { 1 }{ x }$

${ Sin }(-\theta )\quad =\quad -Sin\theta$

${ Cos }(-\theta )\quad =\quad Cos\theta$

${ Tan }(-\theta )\quad =\quad -Tan\theta$

${ Cot }(-\theta )\quad =\quad -Cot\theta$

${ Sec }(-\theta )\quad =\quad Sec\theta$

${ Cosec }(-\theta )\quad =\quad -Cosec\theta$

${ Sin }^{ -1 }(-x)\quad =\quad { Sin }^{ -1 }x$

${ Cos }^{ -1 }(-x)\quad =\quad { \pi -Cos }^{ -1 }x$

$Tan^{ -1 }(-x)\quad =\quad { -Tan }^{ -1 }x$

$Cot^{ -1 }(-x)\quad =\quad { \pi -Cot }^{ -1 }x$

$Sec^{ -1 }(-x)\quad =\quad { \pi -Sec }^{ -1 }x$

$Cosec^{ -1 }(-x)\quad =\quad { -Cosec }^{ -1 }x$

Inverse Function	Domain	Range
Sin^-1x	[-1,1]	$\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right]$
Cos^-1x	[-1,1]	$\left[ 0,\pi \right]$
Tan^-1x	R	$\left( -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right)$
Cot^-1x	R	$\left( \pi ,0 \right)$
Sec^-1x	R-(-1,1)	$\left[ 0,\pi \right] -\left\{ \frac { \pi }{ 2 } \right\}$
Cosec^-1x	R-(-1,1)	$\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right] -\left\{ 0 \right\}$

1.	$Sin(A+B)=SinACosB+CosASinB$	${ Sin }^{ -1 }x+{ Sin }^{ -1 }y={ Sin }^{ -1 }\left[ x\sqrt { 1-{ y }^{ 2 } } +y\sqrt { 1-{ x }^{ 2 } } \right]$
2.	$Sin(A-B)=SinACosB-CosASinB$	${ Sin }^{ -1 }x+{ -Sin }^{ -1 }y={ Sin }^{ -1 }\left[ x\sqrt { 1-{ y }^{ 2 } } -y\sqrt { 1-{ x }^{ 2 } } \right]$
3.	$Sin2A=2SinACosA$	$2Sin^{ -1 }x=Sin^{ -1 }\left[ 2x\sqrt { 1-{ x }^{ 2 } } \right]$
4.	$Sin3A=3SinA-4{ Sin }^{ 3 }A$	$3Sin^{ -1 }x=Sin^{ -1 }\left[ 3x-4x^{ 3 } \right]$
5.	$Cos(A+B)=CosACosB-SinASinB$	$Cos^{ -1 }x+{ Cos }^{ -1 }y= { Cos }^{ -1 }\left[ xy-\sqrt { 1-{ x }^{ 2 } } \sqrt { 1-{ y }^{ 2 } } \right]$
6.	$Cos(A-B)=CosACosB+SinASinB$	$Cos^{ -1 }x-{ Cos }^{ -1 }y={ Cos }^{ -1 }\left[ xy+\sqrt { 1-{ x }^{ 2 } } \sqrt { 1-{ y }^{ 2 } } \right]$
7.	$Cos2A=2Cos^{ 2 }A-1$	$2{ Cos }^{ -1 }x={ Cos }^{ -1 }\left[ 2{ x }^{ 2 }-1 \right]$
8.	$Cos3A=4Cos^{ 3 }A-3CosA$	$3{ Cos }^{ -1 }x={ Cos }^{ -1 }\left[ 4{ x }^{ 3 }-3x \right]$
9.	$Tan(A+B)=\frac { TanA+TanB }{ 1-TanATanB }$	$Tan^{ -1 }x+Tan^{ -1 }y=Tan^{ -1 }\left[ \frac { x+y }{ 1-xy } \right] \quad$
10.	$Tan(A-B) = \frac { TanA-TanB }{ 1+TanATanB }$	$Tan^{ -1 }x-Tan^{ -1 }y=Tan^{ -1 }\quad$
11.	$Tan2A = \frac { 2TanA }{ 1-Tan^{ 2 }A }$	$2Tan^{ -1 }x=Sin^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right)$ $2Tan^{ -1 }x=Cos^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right)$ $2Tan^{ -1 }x=Tan^{ -1 }\left( \frac { 2{ x } }{ 1-{ x }^{ 2 } } \right)$
12.	$Tan3A= \frac { 3TanA-Tan^{ 3 }A }{ 1-3Tan^{ 2 }A }$	$3{ Tan }^{ -1 }x={ Tan }^{ -1 }\left( \frac { 3x-{ x }^{ 3 } }{ 1-3{ x }^{ 2 } } \right)$
13.	${ Tan }^{ -1 }x+{ Tan }^{ -1 }y+{ Tan }^{ -1 }z = Tan^{ -1 }\left( \frac { x+y+z-xyz }{ 1-xy-yz-zx } \right)$	${ Cot }^{ -1 }x+{ Cot }^{ -1 }y=Cot^{ -1 }\left( \frac { xy-1 }{ y+x } \right)$
14.	$Cot(A-B)=\frac { CotACotB+1 }{ CotB-CotA }$	${ Cot }^{ -1 }x-{ Cot }^{ -1 }y=Cot^{ -1 }\left( \frac { xy+1 }{ y-x } \right)$
15.	$Tan(A+B+C) =\frac { TanA+TanB+TanC-TanATanBTanC }{ 1-TanATanB-TanBTanC-TanCTanA }$	${ Tan }^{ -1 }x+{ Tan }^{ -1 }y+{ Tan }^{ -1 }z={ Tan }^{ -1 }\left( \frac { x+y+z-xyz }{ 1-xy-yz-zx } \right)$

Calculus (Differentiation & Integration)

No.	Differentiation	Intergration
1.	$\frac { d }{ dx } { x }^{ n }=n{ x }^{ n-1 },\quad n\neq 0$	$\int { { x }^{ n }dx=\frac { { x }^{ n+1 } }{ n+1 } +c,\quad n\neq -1 }$
2.	$\frac { d }{ dx } { x }=1$	$\int { 1dx=x+c }$
3.	$\frac { d }{ dx } { (c) }=0,$ Where “c” is Constant	$\int { 0dx=c }$ Wher “c” is Constant
4.	$\frac { d }{ dx } { \sqrt { x } }=\frac { 1 }{ 2\sqrt { x } }$	$\int { \frac { 1 }{ \sqrt { x } } dx=2\sqrt { x } } +c$
5.	$\frac { d }{ dx } { \frac { 1 }{ x } }=-\frac { 1 }{ { x }^{ 2 } }$	$\int { \frac { 1 }{ { x }^{ 2 } } dx=\frac { -1 }{ x } } +c$
6.	$\frac { d }{ dx } { \log _{ e }{ x } }=\frac { 1 }{ { x } } ,\quad x\neq 0$	$\int { \frac { 1 }{ x } dx=\log _{ e }{ \left\| x \right\| } } +c,\quad x\neq 0$
7.	$\frac { d }{ dx } \left( { e }^{ x } \right) ={ e }^{ x }$	$\int { { e }^{ x }dx={ e }^{ x } } +c$
8.	$\frac { d }{ dx } { a }^{ x }={ a }^{ x }\log _{ e }{ a }$	$\int { a^{ x }dx=\frac { { a }^{ x } }{ \log _{ e }{ a } } } +c$
9.	$\frac { d }{ dx } (Sinx)=Cosx$ Where “x” is in Radians.	$\int { a^{ x }dx=\frac { { a }^{ x } }{ \log _{ e }{ a } } } +c$ Where “x” is in Radians.
10.	$\frac { d }{ dx } (Cosx)=-Sinx$	$\int { Sinx\quad dx=-Cosx } +c$
11.	$\frac { d }{ dx } (Tanx)=Sec^{ 2 }x$	$\int { { Sec }^{ 2 }x\quad dx=Tanx } +c$
12.	$\frac { d }{ dx } (Cotx)=-Cosec^{ 2 }x$	$\int { { Cosec }^{ 2 }x\quad dx=-Cotx } +c$
13.	$\frac { d }{ dx } (Secx)=SecxTanx$	$\int { Secx\quad Tanx\quad dx=Secx } +c$
14.	$\frac { d }{ dx } (Cosecx)=-CosecxCotx$	$\int { Cosecx\quad Cotx\quad dx=-Cosecx } +c$
15.	$\frac { d }{ dx } \left( Sinh\quad x \right) =Cosh\quad x$ Where “h” is Hyperbolic Funtion Means That : All Graphs of Hyperbolic Functions are Positive.	$\int { Cosh\quad xdx=sinh\quad x+c }$
16.	$\frac { d }{ dx } \left( Cosh\quad x \right) =Sinh\quad x$	$\int { Sinh\quad xdx=Cosh\quad x+c }$
17.	$\frac { d }{ dx } \left( Tanh\quad x \right) =Sech^{ 2 }\quad x$	$\int { Sech^{ 2 }\quad xdx=Tanh\quad x+c }$
18.	$\frac { d }{ dx } \left( Coth\quad x \right) =-Cosech^{ 2 }\quad x$	$\int { Cosech^{ 2 }\quad xdx=-Coth\quad x+c }$
19.	$\frac { d }{ dx } \left( Sech\quad x \right) =-Sech\quad x\quad Tanh\quad x$	$\int { Sech\quad x\quad Tanh\quad x\quad dx=-Sech\quad x+c }$
20.	$\frac { d }{ dx } \left( Cosech\quad x \right) =-Cosech\quad x\quad Coth\quad x$	$\int { Cosech\quad x\quad Coth\quad x\quad dx=-Cosech\quad x+c }$
21.	$\frac { d }{ dx } \left( { Sin }^{ -1 }x \right) =\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } }$	$\int { \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } dx=Sin^{ -1 }x+c,\quad -1\le x\le 1$
22.	$\frac { d }{ dx } \left( { Cos }^{ -1 }x \right) =\frac { -1 }{ \sqrt { 1-{ x }^{ 2 } } }$	$\int { \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } dx=-Cos^{ -1 }x+c,\quad -1\le x\le 1$
23.	$\frac { d }{ dx } \left( { Tan }^{ -1 }x \right) =\frac { 1 }{ 1{ +x }^{ 2 } }$	$\int { \frac { 1 }{ 1+{ x }^{ 2 } } } dx=Tan^{ -1 }x+c$
24.	$\frac { d }{ dx } \left( { Cot }^{ -1 }x \right) =\frac { -1 }{ 1{ +x }^{ 2 } }$	$\int { \frac { 1 }{ 1+{ x }^{ 2 } } } dx=-Cot^{ -1 }x+c$
25.	$\frac { d }{ dx } \left( { Sec }^{ -1 }x \right) =\frac { 1 }{ x\sqrt { { x }^{ 2 }-1 } }$	$\int { \frac { 1 }{ x\sqrt { { x }^{ 2 }-1 } } } dx=Sec^{ -1 }x+c$
26.	$\frac { d }{ dx } \left( Cosec^{ -1 }x \right) =\frac { -1 }{ x\sqrt { { x }^{ 2 }-1 } }$	$\int { \frac { 1 }{ x\sqrt { { x }^{ 2 }-1 } } } dx=-Cosec^{ -1 }x+c$
27.	$\frac { d }{ dx } \left\| x \right\| =\frac { \left\| x \right\| }{ x } ,\quad (x\neq 0)$	$\int { \frac { \left\| x \right\| }{ x } } dx=\left\| x \right\| +c,\quad (x\neq 0)$

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Trig Functions, trigonometric functions, trigonometry table, trigonometry formulas, trigonometry formula, trigonometric formulas, trigonometry formulas list, trigonometric functions, trigonometric formula

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Trig Functions – All Best Math || Trigonometric Funtions 500+

Trigonometric Functions

Basic Trigonometric Functions

Most Useable Trigonometric Functions

Inverse Trigonometric Functions

Calculus (Differentiation & Integration)

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