Trigonometry and Pre-calculus - Division of Academic Enhancement

Formulas from Trigonometry
opp y
=
hyp r
adj x
cosθ=
=
hyp r
opp y sin θ
tan θ=
= =
adj x cosθ
hyp
1
csc θ=
=
opp sin θ
hyp
1
secθ=
=
adj cosθ
adj
1
cosθ
cot θ=
=
=
opp tan θ sin θ
2
2
sin θ+ cos θ=1
2
2
1+ tan θ= sec θ
2
2
1+ cot θ=csc θ
(cos(θ), sin(θ))
sin θ=
Addition Formulas
sin (α+ β)=sin α cos β+ sin βcos α
cos(α+ β)=cos α cosβ−sin α sin β
sin ( α+β) tan α+tan β
tan (α +β)=
=
cos( α+β) 1−tan α tan β
Subtraction Formulas
sin (α−β)=sin α cosβ−sin βcos α
cos(α−β)=cosα cosβ+ sin α sin β
sin (α−β) tan α−tan β
tan (α−β)=
=
cos(α−β) 1+tan α tan β
Half-Angle Formulas
√
√
1−cos α
sin α =±
2
2
1+
cos α
cos α =±
2
2
1−cosα
sin α
tan α =
=
2
sin α
1+cosα
( )
( )
( )
Sine Graph
Quadratic Formula
2
if a x + b x+ c=0 then
−b± √b2 −4a c
x=
2a
−b
−b
vertex =
,f
=(h , k )
2a
2a
( ( ))
Double Angle Formulas
sin 2 α=2 sin α cosα
2
2
cos 2 α=cos α−sin α
2
= 1−2 sin α
= 2 cos2 α−1
2 tan α
tan 2α=
2
1− tan α
2
y= a ( x−h ) + k
Cosine Graph
(x +2 x , y +2 y )
2
1
2
2
2
x − y =( x+ y)( x− y)
2
2
2
x ±2 x y+ y =( x± y)
3
3
2
2
x ± y =( x± y)( x ∓ x y+ y )
2
Tangent Graph
Circular Sector
s= r θ
1
Factoring Formulas
Law of Cosines
2
d = √( x1 −x 2 )2+ ( y 1− y2 )2
midpoint =
sin α sin β sin γ
=
=
a
b
c
2
Distance Formula
Midpoint Formula
Law of Sines
a =b +c −2 b c cosα
2
2
2
b = a + c −2 a c cosβ
2
2
2
c =a + b −2 a b cos γ
Formulas From Algebra
1
A= r 2 θ
2
Constants
π=3.141592653589793238462643
e=2.718281828459045235360287
ϕ=1.618033988749894848204586
Trigonometry and Pre-Calculus Formula Sheet – University of Georgia Division of Academic Enhancement
Associative Property
a+ (b + c)=( a+ b)+ c
a(b c)=(a b) c
Commutative Property
a+ b=b+ a
a b=b a
Distributive Property
a(b+ c)=a b+ a c
Exponents and Radicals
x
y
x+ y
A=l w
Circle
A= π r
f ( x)= x
P=2 l +2 w
2
C=2 π r
Right Circular Cone
1 2
1
V = π r h= Abase h
3
3
2
2
S = π r √r + h
1
x
√x a y=a
Common Function Graphs
Rectangle
()
a x =√ a
Trapezoid
1
A= (a +b) h
2
a ∗a =a
x y
xy
(a ) =a
x
x x
(a b) =a b
x
x
a
a
= x
b
b
x
a
=a x−y
ay
1
a−x = x
a
f ( x)= x
Sphere
y
x
4
V = π r3
3
Line Equations
y 2− y1
x 2− x 1
y=m x+ b
y− y1= m( x− x1 )
S=4π r
2
2
Rectangular Box
m=
V =l w h
S = 2 h l +2 l w+ 2 h w
Right Circular Cylinder
Circle Equations
Center=(h, k) radius=r :
2
2
2
( x−h) + ( y−k ) =r
2
2
y= k± √r −( x−h)
Exponentials and Logarithms
y
y=log b x ⇔ b = x
log( x y)=log ( x)+ log( y)
x
log
=log x−log y
y
a
log (x )=a log( x)
()
2
V =π r h
S = 2 π r h +2 π r
2
Circular Sector
s= r θ
f ( x)= x
3
1
A= r 2 θ
2
Parallelogram
A= bh= absin (θ)
Inverse Trigonometry Functions
f ( x)=arcsin ( x)
loga ( x )
a
=x
x
log a (a )= x
log(1)=0
ln (e)=1
log a (a)=1
log (x)=log 10( x)
ln ( x)=log e ( x)
log( x) ln ( x)
log b ( x)=
=
log(b) ln (b)
f ( x)=∣x∣
f ( x)=arccos( x)
Function Modifications
g( x)=a∗f (b∗x−c )+d
a= vertical stretch
b= horizontal squeeze
c=horizontal shift
d=vertical shift
f ( x)= √x
Formulas from Geometry
Right Triangle
2
2
c =a +b
2
1
A= a b
2
f ( x)=arctan ( x)
Equilateral Triangle
√3 s
√3 s 2
h=
A=
2
4
Triangle
1
1
A= b h= a b sin (θ)
2
2
Trigonometry and Pre-Calculus Formula Sheet – University of Georgia Division of Academic Enhancement