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Trigonometry Cheat Sheet (DRAFT) by

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Quadratic Equation

Discri­minant

b²-4ac > 0
2 real roots
b²-4ac = 0
1 repeated root
b²-4ac < 0
2 imaginary roots

Supple­mentary & Comple­mentary Angles

supple­mentary angles
add up to 180°
comple­mentary angles
add up to 90°

Pythag­orean Theorem

a²+b²=c²

45-45-90 triangles

a=b
c=a√2 or b√2
c/√2=a or b

30-60-90 triangles

c=2b
a=b√3
when a triangle's 3 angles are 30°, 60°, and 90°
a = long leg
b = short leg
c = hypotenuse

Area of a triangle

AΔ = (1/2)bh

SOH-CA­H-TOA

sine
opposi­te/­hyp­otenuse
cosine
adjace­nt/­hyp­otenuse
tangent
opposi­te/­adj­acent

csc, sec, cot - the opposite of SOH-CA­H-TOA

cosecant
hypote­nus­e/o­pposite
secant
hypote­nus­e/a­djacent
cotangent
adjace­nt/­opp­osite
 

Circle Measur­ements

area
πr²
diameter
2r
circum­ference
2πr OR πd
arc length
θr
r = radius
d = diameter
θ = angle

Degrees & Radians

degrees to radians
θ·π/180
radians to degrees
θ·180/π

Unit Circle

Unit Circle values

sin
y
cos
x
tan
y/x
csc
1/y
sec
1/x
cot
x/y
for tan & cot, only use the tops of the fractions

Coterminal Angles

θ ± 360°
θ ± 2π

y = a (sin) b (x - c) + d

a
amplitude change
b
period change
c
horizontal change: + = left, - = right
d
vertical change: + = up, - + down
-sin(x)
reflection across x-axis
sin(-x)
reflection across y=axis

Amplitude & Period

amplitude
vertical
period
horizontal
both always positive
tan, cos, sec, cot: no amplitude
  

Inverses

sin⁻¹(y) = x
sin(x) = y
csc(x) = 1/sin(x)

restri­cting range: usually I & IV
except in cos⁻¹: I & II

Reciprocal Identities

sinθ = 1/cscθ
cscθ = 1/sinθ
cosθ = 1/secθ
secθ = 1/cosθ
tanθ = 1/cotθ
cotθ = 1/tanθ

Ratio Identities

tanθ = sinθ/cosθ
cotθ = cosθ/sinθ

Pythag­orean Identities

sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cos²θ = csc²θ

Double­-Angle Identities

sin2θ = 2sinθcosθ
tan2θ = 2tanθ / 1-tan²θ
cos2θ = cos²θ - sin²θ
cos2θ = 1 - 2sin²θ
cos2θ = 2cos²θ - 1

some more identities

sin(a+b) = sin(a)­cos(b) + cos(a)­sin(b)
sin(a-b) = sin(a)­cos(b) - cos(a)­sin(b)
cos(a+b) = cos(a)­cos(b) - sin(a)­sin(b)
cos(a-b) = cos(a)­cos(b) + sin(a)­sin(b)
tan(a+b) = tan(a)­+tan(b) / 1-tan(­a)t­an(b)
tan(a-b) = tan(a)­+tan(b) / 1+tan(­a)t­an(b)