1-cosx 1 Cosx 3 Find the Solutions

Trigonometric Identities and Formulas

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

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 ,

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
Special Triangles
Special triangles may be used to find trigonometric functions of special angles: 30, 45 and 60 degress.
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
a² = b² + c² - 2 b c cos A
b² = a² + c² - 2 a c cos B
c² = a² + b² - 2 a b cos C

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
Cofunctions Identities
sin(π/2 - X) = cosX
cos(π/2 - X) = sinX
tan(π/2 - X) = cotX
cot(π/2 - X) = tanX
sec(π/2 - X) = cscX
csc(π/2 - X) = secX
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]
Sum to Product Formulas
cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
Difference to Product Formulas
cosX - cosY = - 2sin[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]
sinX - sinY = 2cos[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]
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) ]
Difference of Squares Formulas
sin²X - sin²Y = sin(X + Y)sin(X - Y)
cos²X - cos²Y = - sin(X + Y)sin(X - Y)
cos²X - sin²Y = cos(X + Y)cos(X - Y)

Double Angle Formulas
sin(2X) = 2 sinX cosX
cos(2X) = 1 - 2sin²X = 2cos²X - 1
tan(2X) = 2tanX / [ 1 - tan²X ]
Multiple Angle Formulas
sin(3X) = 3sinX - 4sin³X
cos(3X) = 4cos³X - 3cosX
sin(4X) = 4sinXcosX - 8sin³XcosX
cos(4X) = 8cos⁴X - 8cos²X + 1
Half Angle Formulas
sin (X/2) = + or - √ ( (1 - cosX) / 2 )
cos (X/2) = + or - √ ( (1 + cosX) / 2 )
tan (X/2) = + or - √ ( (1 - cosX) / (1 + cosX) )
= sinX / (1 + cosX) = (1 - cosX) / sinX
Power Reducing Formulas
sin²X = 1/2 - (1/2)cos(2X))
cos²X = 1/2 + (1/2)cos(2X))
sin³X = (3/4)sinX - (1/4)sin(3X)
cos³X = (3/4)cosX + (1/4)cos(3X)
sin⁴X = (3/8) - (1/2)cos(2X) + (1/8)cos(4X)
cos⁴X = (3/8) + (1/2)cos(2X) + (1/8)cos(4X)
sin⁵X = (5/8)sinX - (5/16)sin(3X) + (1/16)sin(5X)
cos⁵X = (5/8)cosX + (5/16)cos(3X) + (1/16)cos(5X)
sin⁶X = 5/16 - (15/32)cos(2X) + (6/32)cos(4X) - (1/32)cos(6X)
cos⁶X = 5/16 + (15/32)cos(2X) + (6/32)cos(4X) + (1/32)cos(6X)
Trigonometric Functions Periodicity
sin (X + 2π) = sin X , period 2π
cos (X + 2π) = cos X , period 2π
sec (X + 2π) = sec X , period 2π
csc (X + 2π) = csc X , period 2π
tan (X + π) = tan X , period π
cot (X + π) = cot X , period π
Trigonometric Tables.
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.