Trigonometric Identities and Formulas

Below are some of the most important definitions, identities and formulas in trigonometry.

  1. Trigonometric Functions of Acute Angles

    sin X = opp / hyp = a / c , csc X = hyp / opp = c / a
    tan X = opp / adj = a / b , cot X = adj / opp = b / a
    cos X = adj / hyp = b / c , sec X = hyp / adj = c / b ,
    acute angle trigonometric functions.

  2. Trigonometric Functions of Arbitrary Angles

    sin X = b / r , csc X = r / b
    tan X = b / a , cot X = a / b
    cos X = a / r , sec X = r / a
    acute angle trigonometric functions.

  3. Special Triangles

    Special triangles may be used to find trigonometric functions of special angles: 30, 45 and 60 degress.
    special triangles.

  4. Sine and Cosine Laws in Triangles

    In any triangle we have:
    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
    triangles.

  5. Relations Between Trigonometric Functions

    cscX = 1 / sinX
    sinX = 1 / cscX
    secX = 1 / cosX
    cosX = 1 / secX
    tanX = 1 / cotX
    cotX = 1 / tanX
    tanX = sinX / cosX
    cotX = cosX / sinX

  6. Pythagorean Identities

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

  7. 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

  8. 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

  9. 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
    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 ] / [ cotY - cotX]

  10. Sum to Product Formulas

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

  11. Difference to Product Formulas

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

  12. 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) ]

  13. 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)

  14. Double Angle Formulas

    sin(2X) = 2 sinX cosX
    cos(2X) = 1 - 2sin 2X = 2cos 2X - 1
    tan(2X) = 2tanX / [ 1 - tan 2X ]

  15. 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

  16. 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

  17. 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)

  18. 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
    tan (X + Pi) = tan X , period Pi
    cot (X + Pi) = cot X , period Pi

  19. Trigonometric Tables.

  20. Properties of The Six Trigonometric Functions. Graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points of each of the 6 trigonometric functions.


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Updated: 22 July 2018 (A Dendane)