This document provides formulas and identities for trigonometric functions including definitions of basic trig functions of acute angles, special right triangles, sine and cosine laws, relationships between trig functions, Pythagorean and negative angle identities, addition formulas, sum and difference formulas, double and multiple angle formulas, half angle formulas, power reducing formulas, and periodicity. Graphs of the six trig functions are also presented.

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
2. Special Triangles
Special triangles may be used to find trigonometric functions of special
angles: 30, 45 and 60 degress.
3. Sine and Cosine Laws in Triangles
3.1 - The sine law
sin A/a = sin B/b = sin C/c
3.2 - The cosine laws

2. a 2 = b 2 + c 2 - 2bc cos A
b 2 = a 2 + c 2 - 2ac cos B
c 2 = a 2 + b 2 - 2ab cos C
4. 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
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

3. 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
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

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

5. 17. Graphs:
17.1. Sine Function : f(x) = sin (x)
17.2. Cosine Function : f(x) = cos (x)
17.3. Tangent Function : f(x) = tan (x)

6. 17.4. Cotangent Function : f(x) = cot (x)
17.5. Secant Function : f(x) = sec (x)
17.6. Cosecant Function : f(x) = csc (x)