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Maths 3, 4: Trigonometry
Identities & Formulas - 1
 1. Trigonometric Functions of Acute Angles


   sin X = a / c
   csc X = c / a        Important!
   tan X = a / b
   cot X = b / a
   cos X = b / c
   sec X = c / b




 2. Special Triangles
Maths 3, 4: Trigonometry
Identities & Formulas - 2
   Special triangles may be used to find trigonometric functions of
   special angles: 30, 45 and 60 degress.




 3. Sine and Cosine Laws in Triangles


   1 - The sine law
   sin A / a = sin B / b = sin C / c
   2 - The cosine laws
   a 2 = b 2 + c 2 - 2 b c cos A
   b 2 = a 2 + c 2 - 2 a c cos B
   c 2 = a 2 + b 2 - 2 a b cos C




 4. Relations Between Trigonometric Functions

   cscX = 1 / sinX, sinX = 1 / cscX
   secX = 1 / cosX, cosX = 1 / secX
Maths 3, 4: Trigonometry
Identities & Formulas - 3
   tanX = 1 / cotX, cotX = 1 / tanX
   tanX = sinX / cosX, cotX = cosX / sinX

 5. Pythagorean Identities

   sin 2X + cos 2X = 1
   1 + tan 2X = sec 2X
   1 + cot 2X = csc 2X

 6. Negative Angle Identities

   sin(-X) = - sinX , odd function
   csc(-X) = - cscX , odd function
   cos(-X) = cosX , even function
   sec(-X) = secX , even function
   tan(-X) = - tanX , odd function
   cot(-X) = - cotX , odd function

 7. Cofunctions Identities

   sin(pi/2 - X) = cosX
   cos(pi/2 - X) = sinX
   tan(pi/2 - X) = cotX
   cot(pi/2 - X) = tanX
   sec(pi/2 - X) = cscX
   csc(pi/2 - X) = secX

 8. Addition Formulas

   cos(X + Y) = cosX cosY - sinX sinY
   cos(X - Y) = cosX cosY + sinX sinY
   sin(X + Y) = sinX cosY + cosX sinY
   sin(X - Y) = sinX cosY - cosX sinY
Maths 3, 4: Trigonometry
Identities & Formulas - 4
   tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY]
   tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY]
   cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY]
   cot(X - Y) = [ cotX cotY + 1 ] / [ cotX - cotY]

 9. Sum to Product Formulas

   cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
   sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]

 10.Difference to Product Formulas

   cosX - cosY = - 2sin[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]
   sinX - sinY = 2cos[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]

 11.Product to Sum/Difference Formulas

   cosX cosY = (1/2) [ cos (X - Y) + cos (X + Y) ]
   sinX cosY = (1/2) [ sin (X + Y) + sin (X - Y) ]
   cosX sinY = (1/2) [ sin (X + Y) - sin[ (X - Y) ]
   sinX sinY = (1/2) [ cos (X - Y) - cos (X + Y) ]

 12.Difference of Squares Formulas

   sin 2X - sin 2Y = sin(X + Y)sin(X - Y)
   cos 2X - cos 2Y = - sin(X + Y)sin(X - Y)
   cos 2X - sin 2Y = cos(X + Y)cos(X - Y)

 13.Double Angle Formulas

   sin(2X) = 2 sinX cosX
   cos(2X) = 1 - 2sin 2X = 2cos 2X - 1
   tan(2X) = 2tanX / [ 1 - tan 2X ]
Maths 3, 4: Trigonometry
Identities & Formulas - 5
 14.Multiple Angle Formulas

   sin(3X) = 3sinX - 4sin 3X
   cos(3X) = 4cos 3X - 3cosX
   sin(4X) = 4sinXcosX - 8sin 3XcosX
   cos(4X) = 8cos 4X - 8cos 2X + 1

 15.Half Angle Formulas

   sin (X/2) = + or - SQRT [ (1 - cosX) / 2 ]
   cos (X/2) = + or - SQRT [ (1 + cosX) / 2 ]
   tan (X/2) = + or - SQRT [ (1 - cosX) / (1 - cosX) ]
   = sinX / (1 + cosX) = (1 - cosX) / sinX

 16.Power Reducing Formulas

   sin 2X = 1/2 - (1/2)cos(2X))
   cos 2X = 1/2 + (1/2)cos(2X))
   sin 3X = (3/4)sinX - (1/4)sin(3X)
   cos 3X = (3/4)cosX + (1/4)cos(3X)
   sin 4X = (3/8) - (1/2)cos(2X) + (1/8)cos(4X)
   cos 4X = (3/8) + (1/2)cos(2X) + (1/8)cos(4X)
   sin 5X = (5/8)sinX - (5/16)sin(3X) + (1/16)sin(5X)
   cos 5X = (5/8)cosX + (5/16)cos(3X) + (1/16)cos(5X)
   sin 6X = 5/16 - (15/32)cos(2X) + (6/32)cos(4X) - (1/32)cos(6X)
   cos 6X = 5/16 + (15/32)cos(2X) + (6/32)cos(4X) + (1/32)cos(6X)

 17.Trigonometric Functions Periodicity

   sin (X + 2Pi) = sin X , period 2Pi
   cos (X + 2Pi) = cos X , period 2Pi
   sec (X + 2Pi) = sec X , period 2Pi
   csc (X + 2Pi) = csc X , period 2Pi
Maths 3, 4: Trigonometry
Identities & Formulas - 6
     tan (X + Pi) = tan X , period Pi
     cot (X + Pi) = cot X , period Pi

18. Graphs of The Six Trigonometric Functions.

Sine Function : f(x) = sin (x)




Cosine Function : f(x) = cos (x)




Tangent Function : f(x) = tan (x)
Maths 3, 4: Trigonometry
Identities & Formulas - 7




Cotangent Function : f(x) = cot (x)




Secant Function : f(x) = sec (x)
Maths 3, 4: Trigonometry
Identities & Formulas - 8




Cosecant Function : f(x) = csc (x)
Maths 3, 4: Trigonometry
Identities & Formulas - 9

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Math34 Trigonometric Formulas

  • 1. Maths 3, 4: Trigonometry Identities & Formulas - 1 1. Trigonometric Functions of Acute Angles sin X = a / c csc X = c / a Important! tan X = a / b cot X = b / a cos X = b / c sec X = c / b 2. Special Triangles
  • 2. Maths 3, 4: Trigonometry Identities & Formulas - 2 Special triangles may be used to find trigonometric functions of special angles: 30, 45 and 60 degress. 3. Sine and Cosine Laws in Triangles 1 - The sine law sin A / a = sin B / b = sin C / c 2 - The cosine laws a 2 = b 2 + c 2 - 2 b c cos A b 2 = a 2 + c 2 - 2 a c cos B c 2 = a 2 + b 2 - 2 a b cos C 4. Relations Between Trigonometric Functions cscX = 1 / sinX, sinX = 1 / cscX secX = 1 / cosX, cosX = 1 / secX
  • 3. Maths 3, 4: Trigonometry Identities & Formulas - 3 tanX = 1 / cotX, cotX = 1 / tanX tanX = sinX / cosX, cotX = cosX / sinX 5. Pythagorean Identities sin 2X + cos 2X = 1 1 + tan 2X = sec 2X 1 + cot 2X = csc 2X 6. Negative Angle Identities sin(-X) = - sinX , odd function csc(-X) = - cscX , odd function cos(-X) = cosX , even function sec(-X) = secX , even function tan(-X) = - tanX , odd function cot(-X) = - cotX , odd function 7. Cofunctions Identities sin(pi/2 - X) = cosX cos(pi/2 - X) = sinX tan(pi/2 - X) = cotX cot(pi/2 - X) = tanX sec(pi/2 - X) = cscX csc(pi/2 - X) = secX 8. Addition Formulas cos(X + Y) = cosX cosY - sinX sinY cos(X - Y) = cosX cosY + sinX sinY sin(X + Y) = sinX cosY + cosX sinY sin(X - Y) = sinX cosY - cosX sinY
  • 4. Maths 3, 4: Trigonometry Identities & Formulas - 4 tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY] tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY] cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY] cot(X - Y) = [ cotX cotY + 1 ] / [ cotX - cotY] 9. Sum to Product Formulas cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ] sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ] 10.Difference to Product Formulas cosX - cosY = - 2sin[ (X + Y) / 2 ] sin[ (X - Y) / 2 ] sinX - sinY = 2cos[ (X + Y) / 2 ] sin[ (X - Y) / 2 ] 11.Product to Sum/Difference Formulas cosX cosY = (1/2) [ cos (X - Y) + cos (X + Y) ] sinX cosY = (1/2) [ sin (X + Y) + sin (X - Y) ] cosX sinY = (1/2) [ sin (X + Y) - sin[ (X - Y) ] sinX sinY = (1/2) [ cos (X - Y) - cos (X + Y) ] 12.Difference of Squares Formulas sin 2X - sin 2Y = sin(X + Y)sin(X - Y) cos 2X - cos 2Y = - sin(X + Y)sin(X - Y) cos 2X - sin 2Y = cos(X + Y)cos(X - Y) 13.Double Angle Formulas sin(2X) = 2 sinX cosX cos(2X) = 1 - 2sin 2X = 2cos 2X - 1 tan(2X) = 2tanX / [ 1 - tan 2X ]
  • 5. Maths 3, 4: Trigonometry Identities & Formulas - 5 14.Multiple Angle Formulas sin(3X) = 3sinX - 4sin 3X cos(3X) = 4cos 3X - 3cosX sin(4X) = 4sinXcosX - 8sin 3XcosX cos(4X) = 8cos 4X - 8cos 2X + 1 15.Half Angle Formulas sin (X/2) = + or - SQRT [ (1 - cosX) / 2 ] cos (X/2) = + or - SQRT [ (1 + cosX) / 2 ] tan (X/2) = + or - SQRT [ (1 - cosX) / (1 - cosX) ] = sinX / (1 + cosX) = (1 - cosX) / sinX 16.Power Reducing Formulas sin 2X = 1/2 - (1/2)cos(2X)) cos 2X = 1/2 + (1/2)cos(2X)) sin 3X = (3/4)sinX - (1/4)sin(3X) cos 3X = (3/4)cosX + (1/4)cos(3X) sin 4X = (3/8) - (1/2)cos(2X) + (1/8)cos(4X) cos 4X = (3/8) + (1/2)cos(2X) + (1/8)cos(4X) sin 5X = (5/8)sinX - (5/16)sin(3X) + (1/16)sin(5X) cos 5X = (5/8)cosX + (5/16)cos(3X) + (1/16)cos(5X) sin 6X = 5/16 - (15/32)cos(2X) + (6/32)cos(4X) - (1/32)cos(6X) cos 6X = 5/16 + (15/32)cos(2X) + (6/32)cos(4X) + (1/32)cos(6X) 17.Trigonometric Functions Periodicity sin (X + 2Pi) = sin X , period 2Pi cos (X + 2Pi) = cos X , period 2Pi sec (X + 2Pi) = sec X , period 2Pi csc (X + 2Pi) = csc X , period 2Pi
  • 6. Maths 3, 4: Trigonometry Identities & Formulas - 6 tan (X + Pi) = tan X , period Pi cot (X + Pi) = cot X , period Pi 18. Graphs of The Six Trigonometric Functions. Sine Function : f(x) = sin (x) Cosine Function : f(x) = cos (x) Tangent Function : f(x) = tan (x)
  • 7. Maths 3, 4: Trigonometry Identities & Formulas - 7 Cotangent Function : f(x) = cot (x) Secant Function : f(x) = sec (x)
  • 8. Maths 3, 4: Trigonometry Identities & Formulas - 8 Cosecant Function : f(x) = csc (x)
  • 9. Maths 3, 4: Trigonometry Identities & Formulas - 9