Math34 Trigonometric Formulas

1. Maths 3/4: Trigonometry
Formulas & Identities
1. Trigonometric Functions of Acute Angles
sin X = a / c
csc X = c / a Basic
tan X = a / b
cot X = b / a
cos X = b / c
sec X = c / b
1. Special Triangles
Special triangles may be used to find trigonometric functions of
special angles: 30, 45 and 60 degress.
2. Sine and Cosine Laws in Triangles

2. 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
3. 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
4. Pythagorean Identities
sin 2X + cos 2X = 1
1 + tan 2X = sec 2X
1 + cot 2X = csc 2X
5. Negative Angle Identities
sin(-X) = - sinX , odd function
csc(-X) = - cscX , odd function
cos(-X) = cosX , even function
sec(-X) = secX , even function

3. tan(-X) = - tanX , odd function
cot(-X) = - cotX , odd function
6. 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
7. 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 ] / [ cotX - cotY]
8. Sum to Product Formulas
cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
9. Difference to Product Formulas
cosX - cosY = - 2sin[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]
sinX - sinY = 2cos[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]
10.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) ]

4. 11.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)
12.Double Angle Formulas
sin(2X) = 2 sinX cosX
cos(2X) = 1 - 2sin 2X = 2cos 2X - 1
tan(2X) = 2tanX / [ 1 - tan 2X ]
13.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
14.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
15.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)

5. 16.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
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)

6. Cotangent Function : f(x) = cot (x)
Secant Function : f(x) = sec (x)

7. Cosecant Function : f(x) = csc (x)