TRIGONOMETRY IDENTITIES

 TRIGONOMETRY IDENTITIES

Calculation of trigonometric ratios for specific angles establish some identities are called Trigonometry identities. Or can be say that an equation involving trigonometric ratios of an angle is called a trigonometry identity.

• SIN A = 1/COSEC A
• COS A = 1/SEC A
• TAN A = 1/COT A    OR      SIN A / COS A 
• COT A = 1/TAN A    OR      COS A / SIN A
• SEC A = 1/COS A 
• COSEC A = 1/SIN A
• COS² A+ SIN²A = 1 OR SIN²A + COS²A =1
• SEC²A  - TAN²A = 1  OR   1+ TAN²A = SEC² A 
• 1+COT² A= COSEC² AOR COSEC²A-COT² A=1
  
  
  
  

ADDITION AND SUBTRACTION

Ø  SIN (A+B) = SINACOSB +COSASINB
Ø  SIN (A-B) = SINACOSB – COSASINB
Ø  COS (A+B) = COSACOSB – SINASINB
Ø  COS (A-B) = COSACOSB + SINASINB
Ø  TAN (A+B) =TANA + TANB/1-TANATANB
Ø  TAN (A-B) = TANA-TANB/1 +TANATANB
Ø  COT (A-B) = 1-COTACOTB/COTA+COTB
Ø  COT (A-B) = 1+COTACOTB/COTA-COTB

 C & D FORMULA 

Ø  SINC + SIND = 2SIN(C+D/2).COS(C-D/2)
Ø  SINC – SIND =2COS(C +D/2).SIN(C-D/2)
Ø  COSC +COSD = 2COS(C+D/2).COS(C-D/2)
Ø  COSC-COSD = 2COS(C+D/2).COS(C-D/2)

MULTIPLICATION

A.    -2SINASINB = COS(A+B)-COS(A-B)

                                                 OR
                               = COS(A-B)-COS(A+B)
                                                  OR  
                               = -COS(A+B)+COS(A+B)
B.     2SINACOSB =SIN(A+B)+SIN(A-B)
C.    2COSASINB=SIN(A+B)-SIN(A-B)
D.    2COSACOSB=COS(A+B)+COS(A-B)

MULTIPLES

 I. SIN2A = 2SINACOSA
                    = 2TANA/1+TAN2A
 II. COS2A = COS²A-SIN²A
                     =1-TAN²A/1+TAN²A
                     =1-2SIN²A
                     =2COS²A-1
III. TAN2A=2TANA/1-TAN²A
IV. SIN3A = 3SINA-4SIN³A
 V. COS3A = 4COS³A-3SINA
 VI.TAN3A=3TANA-TAN³A/1-3TANA
VII. 1+COS2A=2COS²A
VIII. 1-COS2A=2SIN²A

SUB-MULTIPLES

• SINA = 2SINA/2COSA/2
•                     = 2TANA/2/1+TAN²A/2
• 
  
• COSA = COS²A/2-SIN²A/2
•                      =1-TAN²A/2/1+TAN²A/2
•                      =1-2SIN²A/2
•                      =2COS²A/2-1
• 
  
• TANA=2TANA/2/1-TAN²A/2
• 
  
• 1+COSA=2COS²A/2
• 
  
• 1-COSA=2SIN²A/2
• TANπ/4+X=1+TANX/1-TANX
• TANπ/4-X=1-TANX/1+TANX      

• Inverse Trigonometric Formulas

• Inverse Trigonometric Formulas: Trigonometry is a part of geometry, where we learn about the relationships between angles and sides of a right-angled triangle. In Class 11 and 12 Maths syllabus, you will come across a list of trigonometry formulas, based on the functions and ratios such as, sin, cos and tan. Similarly, we have learned about inverse trigonometry concepts also. The inverse trigonometric functions are written as sin-1x, cos-1x, cot-1 x, tan-1 x, cosec-1 x, sec-1 x. Now, let us get the formulas related to these functions.

  What is Inverse Trigonometric Function?

  The inverse trigonometric functions are also known as the anti trigonometric functions or sometimes called arcus functions or cyclometric functions. The inverse trigonometric functions of sine, cosine, tangent, cosecant, secant and cotangent are used to find the angle of a triangle from any of the trigonometric functions. It is widely used in many fields like geometry, engineering, physics, etc. But in most of the time, the convention symbol to represent the inverse trigonometric function using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). To determine the sides of a triangle when the remaining side lengths are known.

  Consider, the function y = f(x), and x = g(y) then the inverse function is written as g = f-1,

  This means that if y=f(x), then x = f-1(y).

  Such that f(g(y))=y and g(f(y))=x.

  Example of Inverse trigonometric functions: x= sin-1y

  The list of inverse trigonometric functions with domain and range value is given below:

  Functions	Domain	Range
  Sin-1 x	[-1, 1]	[-π/2, π/2]
  Cos-1x	[-1, 1]	[0, π/2]
  Tan-1 x	R	(-π/2, π/2)
  Cosec-1 x	R-(-1,1)	[-π/2, π/2]
  Sec-1 x	R-(-1,1)	[0,π]-{ π/2}
  Cot-1 x	R	[-π/2, π/2]-{0}

  Inverse Trigonometric Formulas List

  

  
  

  To solve the different types of inverse trigonometric functions, inverse trigonometry formulas are derived from some basic properties of trigonometry. The formula list is given below for reference to solve the problems.

  S.No	Inverse Trigonometric Formulas
  1	sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
  2	cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
  3	tan-1(-x) = -tan-1(x), x ∈ R
  4	cosec-1(-x) = -cosec-1(x), |x| ≥ 1
  5	sec-1(-x) = π -sec-1(x), |x| ≥ 1
  6	cot-1(-x) = π – cot-1(x), x ∈ R
  7	sin-1x + cos-1x = π/2 , x ∈ [-1, 1]
  8	tan-1x + cot-1x = π/2 , x ∈ R
  9	sec-1x + cosec-1x = π/2 ,|x| ≥ 1
  10	sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1
  11	cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1
  12	tan-1(1/x) = cot-1(x), x > 0
  13	tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1
  14	tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1
  15	2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1
  16	2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0
  17	2tan-1 x = tan-1(2x/(1-x2)), -1<x<1
  18	3sin-1x = sin-1(3x-4x3)
  19	3cos-1x = cos-1(4x3-3x)
  20	3tan-1x = tan-1((3x-x3)/(1-3x2))
  21	sin(sin-1(x)) = x, -1≤ x ≤1
  22	cos(cos-1(x)) = x, -1≤ x ≤1
  23	tan(tan-1(x)) = x, – ∞ < x < ∞.
  24	cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞
  25	sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞
  26	cot(cot-1(x)) = x, – ∞ < x < ∞.
  27	sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2
  28	cos-1(cos θ) = θ, 0 ≤ θ ≤ π
  29	tan-1(tan θ) = θ, -π/2 < θ < π/2
  30	cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2
  31	sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π
  32	cot-1(cot θ) = θ, 0 < θ < π
  33	${\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2}),ifx,y\ge 0and{x}^2+y^2\le 1$
  34	${\sin }^{-1}x+{\sin }^{-1}y=\pi -{\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})$, if x, y ≥ 0 and x2+y2>1.
  35	${\sin }^{-1}x-{\sin }^{-1}y=\pi -{\sin }^{-1}(x\sqrt{1-y^2}-y\sqrt{1-x^2})$, if x, y ≥ 0 and x2+y2≤1.
  36	${\sin }^{-1}x-{\sin }^{-1}y=\pi -{\sin }^{-1}(x\sqrt{1-y^2}-y\sqrt{1-x^2})$, if x, y ≥ 0 and x2 +y2>1.
  37	${\cos }^{-1}x+{\cos }^{-1}y={\cos }^{-1}(xy-\sqrt{1-x^2}\sqrt{1-y^2})$, if x, y >0 and x2+y2 ≤1.
  38	${\cos }^{-1}x+{\cos }^{-1}y=\pi -{\cos }^{-1}(xy-\sqrt{1-x^2}\sqrt{1-y^2})$, if x, y >0 and x2+y2>1.
  39	${\cos }^{-1}x-{\cos }^{-1}y={\cos }^{-1}(xy+\sqrt{1-x^2}\sqrt{1-y^2})$, if x, y > 0 and x2+y2≤1.
  40	${\cos }^{-1}x-{\cos }^{-1}y=\pi -{\cos }^{-1}(xy+\sqrt{1-x^2}\sqrt{1-y^2})$,if x, y > 0 and x2 +y2>1.