TRIGONOMETRIC

IDENTITIES
RECIPROCAL RELATIONS
1 1 1 1
sinA= cotA= cosA = secA=
cscA tanA secA cosA

1 1
tanA= cscA=
cotA sinA
 EVEN-ODD IDENTITIES
sin (−ϴ )=−sinϴcos (−ϴ ) =cosϴtan (−ϴ ) =−tanϴ
cot (−ϴ )=−cotϴsec (−ϴ ) =secϴcsc (−ϴ )=−cscϴ

COFUNCTION IDENTITIES
sin ( ϴ )=cos ( 90−ϴ ) cosϴ=sin ( 90−ϴ )tanϴ=cot ⁡( 90−ϴ)
cotϴ=tan ⁡(90−ϴ)secϴ=csc ⁡(90−ϴ)cscϴ=sec ⁡(90−ϴ)
PYTHAGOREAN RELATIONS
sin 2 A+ co s 2 A=11+co t 2 A=csc ² A1+tan 2 A=sec ² A

SUM OF ANGLES FORMULAS

 sin ( A+ B )=sinAcosB +cosAsinB

tanA +tanB
cos ( A+ B ) =cosAcosB−sinAsinB tan ( A+ B ) =
1−tanAtanB

DIFFERENCE OF ANGLES
FORMULAS
 sin ( A−B )=sinAcosB −cosAsinB

tanA−tanB
cos ( A−B )=cosAcosB +sinAsinB tan ( A−B )=
1+tanAtanB

DOUBLE ANGLE FORMULAS

sin 2 A=2 sinAcosA cos 2 A=co s 2 A−si n2 A

2 tanA
tan2 A=
1−tan ² A
POWERS OF

FUNCTIONS
 1 1
sin 2 A= (1−cos 2 A )co s2 A= ( 1+cos 2 A )
2 2

1−cos 2 A
ta n2 A=
1+ cos 2 A
FUNCTIONS OF HALF ANGLES

A 1−cosA A 1+ cosA
sin
2
=±
√
2
cos =±
2 √2

A 1−cosA sinA
tan = =
2 sinA 1+cosA
SUM OF TWO ANGLES
1 1
sinA +sinB =2sin ( A+ B ) cos ( A−B)
2 2
 1 1
cosA +cosB=2 cos ( A+ B ) cos ( A−B)
2 2

sin ⁡( A+ B)
tanA +tanB=
cosAcosB

DIFFERENCE OF TWO
FUNCTIONS
 1 1
sinA−sinB=2 cos ( A + B ) sin ( A−B)
2 2

1 1
cosA −cosB=2 sin ( A+ B ) sin ( A−B)
2 2

sin ⁡(A−B)
tanA−tanB=
cosAcosB
PRODUCT OF TWO

FUNCTIONS
2 sinAsinB=cos ( A−B )−cos ( A+ B )
 2 sinAcosB=sin ( A+ B )+ sin ( A−B )

2 cosAcosB=cos ( A+ B ) +cos ⁡( A−B)

ALGEBRA
 QUADRATIC FORMULA
Form: A x 2+ bx+ c=0

−b ± √ b2−4 ac of roots ; x1 + x 2=
−b
Roots ; x=
2a
∑ a
c
Product of roots ; x1 ( x 2 )=
a
BINOMIAL THEOREM
Form : ( x + y ) n

r th term : nCm∗x n−m y m

where: m=r-1
ARITHMETIC PROGRESSION
n
d=a2−¿a =a −a ¿a n=a1 +(n−1)d a n=a x + ( n−x ) d Sn= (a 1+ an )
1 3 2
2
GEOMETRIC PROGRESSION
n−1
a 2 a5 a ( r n−1)
a n=a1 r r= = =… s= 1 when r >1
a1 a 4 r−1
a 1( 1−r n )
s= when r <1
1−r
 INFINITE GEOMETRIC
PROGRESSION
For a geometric progression where 0 < r < 1 and n = ∞
a 1
∑ of I .G . P .= 1−r
HARMONIC PROGRESSION

- reciprocal of arithmetic progression

 SPHERICAL
TRIGONOMETRY
Sine Law:
sina sinb sinb
= =
sinA sinB sinC

Laws of Cosines for sides:

cos=cosbcosc+ sinbsinccosA

Law of Cosines for angles:

cosA =−cosBcosC +sinBsinCcosA
Spherical Polygon:
πR ² E
A= E = spherical excess
180 °
E = (A+B+C+D…) – (n-2)180°

Spherical Pyramid:
1 π R3 E
V = A B H=
3 540°
ANALYTIC
GEOMETRY
DISTANCE BETWEEN
TWO POINTS
2
√
d= ( x 2−x 1 ) +¿ ¿

SLOPE OF A LINE (m)

 rise ∆ y
m= =
run ∆ x

y 2−¿ y
m= 1
¿
x 2−x 1
ANGLE BETWEEN TWO
LINES
m2−m1
tanθ=
1+ m 2 m 1
m −m1
θ=tan −1( 2 )
1+m2 m1
DISTANCE BETWEEN A
POINT AND A LINE
A x 1+ B y 1 +C
d=
± √ A 2+ B 2
 DISTANCE BETWEEN
TWO PARALLEL LINES
C 1−C2
d=
± √ A 2+ B 2

*use the sign that would make the distance positive

COORDINATES OF A POINT
THAT DIVIDES A LINE
SEGMENT
x1 r2 + x2 r1
x= -For abscissa
r 1 +r 2
 y 1 r 2+ y 2 r 1
y= -For ordinate
r 1+ r 2

AREA OF A POLYGON USING

THE COORDINATES OF
VERTICES
 1 x x x x
A= ∨ 1 2 3 1 ∨¿
2 y1 y2 y3 y1
1
A= ¿
2

LINE
General Equation: A x 2+ bx+ c=0
Point-Slope Form: y− y1 =m(x−x 1)
 Point-Intercept Form: y=mx+b
y
Two-Point Form: y− y1 = x −x ( x −x1 )¿
¿

2 1
x y
Intercept Form: + =1
a b

CONIC SECTION
General Equation:
A x 2+ Bxy +C y 2 + Dx+ Ey + F=0
 B2−4 AC Conic Section Eccentricity
<0 Ellipse < 1.0
=0 Parabola = 1.0
>0 Hyperbola > 1.0

CIRCLE
General Equation:
x 2+ y 2+ Dx + Ey+ F=0
*If D & E = 0, center is at the origin (0,0)
*If either D or E, or both D & E ≠ 0, the center is at (h,k)

Standard Equation:

C(0,0)
x 2+ y 2=r 2
C(h,k)
( x−h)2 +( y−k )2=r 2
General Equation:

A x 2+C y 2+ Dx + Ey+ F=0

Center (h,k)
−D −E
h= k=
2A 2A
Radius (r)
 D2 + E 2−4 AF
r=
√ 4A²

PARABOLA
-is the locus of a point which moves so that is it always equidistant to a fixed point
called focus and a fixed straight line called directrix.

General Equation: Elements:

Ax2 + Dx+ Ey + F=0 Eccentricity, e:
 df
Cy 2+ Dx+ Ey+ F=0 e= =1
dd
Standard Equation:
( x−h )2=± 4 a ( y−k ) Length of latus rectums, LR:
( y−k )²=± 4 a(x−h) LR=4 a
CONVERSION OF GENERAL EQUATION
TO STANDARD EQUATION
For axis horizontal: Cy 2+ Dx+ Ey+ F=0
 E 2−4 CF k=
−E
a=
−D
h=
4 CD 2C 4C
 CONVERSION OF GENERAL EQUATION
TO STANDARD EQUATION
For axis vertical: Ax2 + Dx+ Ey + F=0

h=
−D D 2−4 AF a=
−E
k=
2A 4 AE 4A

ELLIPSE
- is a locus of a point which moves so that the sum of its distance to the fixed points (foci) is
constant and is equal to the length of the major axis (2a).

General Equation:
Ax2 +Cy 2+ Dx+ Ey+ F=0
Standard Equation: Ellipse with center at (h, k)

(x−h) ² ( y−k )² (x−h) ² ( y−k )²

+ =1 + =1
a² b² b² a²
Standard Equation: Ellipse with center at origin
x² y ² x² y ²
+ =1 + =1
a² b² b² a²

Elements:

Location of foci, c: Eccentricity, e:

c
c 2=a2−b 2 e=
a
 Length of LR: Location of directrix,

2b² a
LR= d=
a e

HYPERBOLA
- Hyperbola can be defined as the locus of point that moves such that the difference of its
distances from two fixed points called the foci is constant. The constant difference is the
length of the transverse axis, 2a.

General Equation:
Ax ²−Cy ²+ Dx+ Ey+ F=0
Standard Equation:
(x−h) ² ( y −k )² ( y−k )² ( x−h)²
− =1 − =1
a² b² b² a²
Elements:

Location of foci, c:

c 2=a2+ b2
Eq’n of asymptote:
y−k=± m(x−h)
Same as ellipse:
 Length of LR, directrix,
and eccentricity.

(+)m =upward asymptote

(-)m = upward asymptote
m = b/a if transverse axis is
horizontal and vice versa
ENGINEERING
ECONOMY
 SIMPLE INTEREST
I =Prt
F=P(1+rt )

ORDINARY EXACT
 (Banker’s year) (Actual number of
days in year)

1 year = 360 days 1 year = 365 days

1 month = 30 days 1 leap year = 366 days
 REINFORCED
CONCRETE DESIGN

NSCP COEFFICENTS
- Hyperbola can be defined as the locus of point that moves such that the difference of its
distances from two fixed points called the foci is constant