TRIGONOMETRYS U M M A RY O F F I R S T Q UA RT E R L E SS O N S

ARCE REYES VARONA

HISTORYORIGIN AND SCIENTISTS

TRIGONOMETRY

• emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

Greek words Trigōnon = "triangle" Metron = "measure

ORIGIN:Greek words Trigōnon = "triangle" Metron = "measure

Greek words Trigōnon = "triangle" Metron = "measure

Greek words Trigōnon = "triangle" Metron = "measure

• branch of Mathematics that studies relationships involving lengths and angles of triangles

HIPPARCHUS• Father of Trigonometry• first person of whose

systematic use of trigonometry we have documentary evidence

• made astronomical observations from 161 to 127 BC.

RENÉ DESCARTES• Father of Modern

Philosophy• Cartesian Coordinate

System• Cartesian Plane

CONCEPTS

FORMULAE, SAMPLE PROBLEMS

𝜋𝜔±()Δ 𝑦Δ𝑥

CARTESIAN PLANE

DISTANCE FORMULAThe distance between two

points, (x1, y1) and (x2, y2),

is given by the formula

MIDPOINT FORMULAThe midpoint between two

points, (x1, y1) and (x2, y2)

is given by the formula:

UNIT CIRCLEa circle in the Cartesian

plane whose center is the origin, (0,0), and whose

radii are equal to one.

ANGLES

An angle is in its standard position if its vertex is located at the origin and one ray is on

the positive x-axis.

REFERENCE ANGLEsmallest angle that the terminal side of a given angle makes with the x-

axis. Reference angles are less than 90 degrees

IF THE ANGLE IN STANDARD POSITION IS IN

Angle in Standard Position

Reference Angle

QUADRANT 1

IF THE ANGLE IN STANDARD POSITION IS IN

Reference Angle

180 - Angle in Standard Position

QUADRANT 2

IF THE ANGLE IN STANDARD POSITION IS IN

Reference Angle

Angle in Standard Position – 180

QUADRANT 3

IF THE ANGLE IN STANDARD POSITION IS IN

Reference Angle

Angle in Standard Position – 360

QUADRANT 4

COTERMINAL ANGLEStwo angles are coterminal if

they have the same position in the Cartesian plane

DECIMAL DEGREES DECIMAL MINUTES SECONDS

1°= 60’ (minutes) = 3600” (seconds)

1/60°= 1’ (minute) 1/3600°= 1” (second)

Vise versa

EXAMPLE:

DEGREE RADIAN

Radian x 180 π

RADIAN DEGREE

Degree x π 180

EXAMPLE:

CENTRAL ANGLECentral Angle= (s/r) s=length of arc r=radius

ANGULAR SPEED the distance travelled by the body in terms of rotations or revolutions to the time taken

Angular speed (w) =

ANGULAR SPEED (COMPLETE ROATATION)

2π= 360 degrees or one complete

rotation/revolution

The relation between Linear speed and Angular speed is

EXAMPLE:Question 1: Earth takes 365 days to complete a revolution around the sun. Calculate its Angular speed?Solution: Angular Speed is given by ω = D/T Where D = Rotational distance traveled = 2π and T = 365 × 24 × 60 × 60 = 31536000 S. ∴ Angular Speed (ω) = 2π/31536000 = 1.9923 × 10-7 rad/s.

EXAMPLE:Question 2: 

The wheel of bullock cart of radius 1m is moving with the speed of 5m per second. Calculate its Angular speed.Solution: Given: Linear speed V = 5m/s, Radius of Circular path r = 1m The Angular Speed ω = V/r                                             = (5m/s)/1m                                             = 5 rad/s.

LINEAR SPEED

speed with which the body moves in the linear path. In simple words, it is the distance traveled for linear path in

given time

s = distancet = timew = angular speedr = raidus

EXAMPLE:Question 1: A ferris wheel with a radius of 15 in has 3 revolutions per second. Convert to miles per hour.Solution:

EXAMPLE:Question 2: A ferris wheel with a radius of 10 ft makes 6 revolutions in 1 sec. Convert to inches per min.Solution:

Asector=1(r2 θ) 2

Asegment=Asec=A 

Arc Length= rθ

TRIGONOMETRIC FUNCTIONS

r= distance of (x,y) from the originr=√(x2+y2) sin θ=y/r tan θ=y/x sec θ= r/x (x ≠0)cos θ=x/r csc θ=r/y (y ≠0) cot θ= x/y (y ≠0) 

ENDS U M M A RY O F F I R S T Q UA RT E R L E SS O N S

ARCE REYES VARONA