1 Sector Sector area= r Θ 2
Angul Angular ar speed speed =Θ time r Θ Linea Linearr speed speed = time Arc length length= r Θ
opp hyp csc = hyp opp adj hyp sec = cos = hyp adj opp adj tan = cot = adj opp Deg/Rad convert D ∗Π 180O R= D = O Π 180 sin =
Law of Sines (AAS, ASA) Law of cosines (SSS, sin A sin B sin C SAS) = = a b c a 2=b 2+ c 2−2 bc co cosA sA Area for SAS A =0.5 bc sin A
Area for AAS, ASA 2 sin B sin C A =0.5 a sin A
Area =√ ( s ( s − a )( s −b )( s − c )) Area for SSS: 1 s = ( a +b +c ) 2 Graphing trig functions: Ambiguous case If a is acute: a b a h y = a sin (bx +c )+ d (SSA): =No solution h =b sin A a ≠0 ; b ≠ 0 a =opp Amplitude: ∣a∣ a b a =h b =adj 2Π Period: =One solution b If a is obtuse: 1
Frequency:
a ≤b
= No solution
period c Phase shift: − ∣b∣ Midline: y Midline: y = d
a b
a b
a h
2
2
Θ+ cos Θ= 1
= One solution
cos Θ = sin
Π −Θ
( ) ( Π −Θ) 2
2
csc (−Θ)=−csc Θ
1− cos2 Θ sin Θ= 2
cos
2
2
1 cos 2
2
(√ −+ ΘΘ )
tan Θ =± 2
1 cos 1 cos
1−cos Θ tan Θ = 2 sin Θ sin Θ tan Θ = 2 1+cos Θ
tan
Sum identities 2
2
Θ+1= sec Θ
cot
2
2
Θ +1=csc Θ
tan Θ = cot cot Θ= tan
Π −Θ
( ) ( Π −Θ ) 2
2
cos (−Θ)= cos Θ sec (−Θ)= sec Θ
cos
cos ( a +b )= cos a cos b −sin a sin b sin ( a +b )=sin a cos B +cos a sin B
Π −Θ
( ) tan (a+ b)= 1tan−tana +atantanbb csc Θ= sec ( Π −Θ) Difference identities cos ( a −b )= cos a cos b +sin a sin b sec Θ =csc
2
2
tan (−Θ)=−tan Θ cot (−Θ)=− cot Θ
Power-Reducing Identities 2
1
=One solution
Odd-Even Identities
sin (−Θ)=− sin Θ
cos
(√ − Θ ) Θ =± + Θ ) √(
Θ =±
a ≥b
Confunction identities sin Θ= cos
sin
=Two solutions
Pythagorean identitiess sin
Half-Angle Identities
2
Θ=
sin ( a −b )=sin a cos b −cos a sin b
tan ( a −b )=
tan a −tan b 1 +tan a tan b
Double angle identities 1 +cos cos 2 Θ 2
tan
2
Θ=
1−cos cos 2 Θ
sin sin 2 Θ= 2sin Θ cos Θ
1+cos cos 2 Θ
Binomial probability Prob =nC x p x q n− x n independent trials, p trials, p probability probability of success, q = 1 - p
tan 2 Θ=
2 tan Θ 1− tan
2
Θ
cos 2 Θ=cos
2
2
Θ−sin Θ
cos cos 2 Θ= 2cos
2
Θ−1
cos cos 2 Θ= 1− 2sin
2
Θ
Sigma notation
nth term of arith. seq. Sum of arith. series n a n= a1+( n −1) d S N = ( a 1+a n)
k
a = a +... a ∑ = n
1
2
k
Sum of arith. series n S n= [ 2a 1+( n−1 ) d ]
nth term of geo. seq. n− 1 a n= a1 r
2
n 1
Sum of finite geo. series
Sum of finite geo. series a1− a n r s n= 1− r
(−) n
S n= a1
1− r 1
r
Sum of inf. geo. series Binomial expansion a1 0,1,2, n (a +b ) n=nC r an−r br wherer =0,1,2,n r < ∣ ∣ 1 S = 1− r B A ≠0 Conic rotation tan 2 Θ= A−C Conic sections
Ellipses
Parabolas
Hyperbolas
l a t n o z i r o H
( x −h )2 ( y − k )2 + =1 2 2 a
( x −h )
2
−
2
( y − k )
2
=1
2
b
Vertex
(h,k)
Center
(h,k)
Focus
(h + FL, k)
Foci
(h ± c, k)
Vertices
(h ± a, k)
y = k Axis of symmetry
Covertices
(h, k ± b)
Directrix x = h – FL
Transverse axis ax is y = k
a
b
Center
(h,k)
Foci
(h ± c, k)
Vertices
(h ± a, k)
Major axis y = k
Conjugate axis x = h
Minor axis x = h
Asymptotes
y − k =
±b x h ( − ) a
Eccentricity (vertical or horizontal): c e= a
l a c i t r e V
( x −h )2 ( y − k )2 + =1 2 2 b
( x − h )2= 4FL ( y − k )
( y − k )
2
2
−
( x − h) 2
2
=1
a
Vertex
(h, k)
Center
(h,k)
Focus
(h, k + FL)
Foci
(h ± c, k)
Vertices
(h ± a, k)
x = h Axis of symmetry
Covertices
(h, k ± b)
Directrix y = k - FL
Transverse axis ax is x = h
Circles
Conjugate axis y = k
Major axis y = k Minor axis x = h C B A
( y − k )2= 4FL ( x −h )
c 2 = a2 − b 2
( x − h )2+( y − k )2= r 2 Points are as same as an ellipse. Divide by r 2 to find a and b. Eccentricity is 0.
a
b
Center
(h, k)
Foci
(h, k ± k ± c)
Vertices
(h, k ± k ± a)
Asymptotes
c 2 = a2 + b 2
y − k =
±a b
( x − h)
