5.3 PROPERTIES OF

TRIGONOMETRIC FUNCTIONS

*Periodicity

*Identify Quadrant where angle exists

PERIODICITY A function whose value repeats at regular

intervals is called a “Periodic Function”

The trigonometric functions are periodic functions

Sine, Cosine, Cosecant, and Secant’s values repeat every 360⁰ (2π radians)

Tangent and Cotangent’s values repeat every 180⁰ (π radians)

PERIODIC PROPERTIES

sin( 360) sin sin( 2 ) sin

cos( 360) cos cos( 2 ) cos

csc( 360) csc csc( 2 ) csc

sec( 360) sec sec( 2 ) sec

PERIODIC PROPERTIES

tan( 180) tan tan( ) tan

cot( 180) cot cot( ) cot

tan( 180) tan

y y

x x

-y

y

-x

x

sin 420

sin(60 360)

sin 60

1

2 33

sin 4202

136sec

6sec( 2 )

6sec

12

3

2 3136 3sec

103tan

73tan( )

103tan 3

73tan

43tan( ) 43tan

3tan( )

3tan

1

2 3

NAME THAT QUADRANT! You can narrow down which quadrant an

angle exists in by simply knowing the sign of the value of one of the six trig functions.

If you know the sign of the value of TWO of the six trig functions, you can state with certainty which quadrant the angle exists in.

EXPLORE siny

r

sin 0 sin 0

sin 0 sin 0

++

- -

“r” is always positive

cscr

y

EXPLORE cosx

r

cos 0 cos 0

cos 0 cos 0

+

+-

-

“r” is always positive

secr

x

EXPLORE tany

x

tan 0 tan 0

tan 0 tan 0

+

-+

-

cotx

y

sin 0, cos 0

I II II III

Quadrant II

tan 0, sec 0

I III II III

Quadrant III

csc 0, cos 0

III IV II III

Quadrant III

cot 0, sec 0

II IV I IV

Quadrant IV

ASSIGNMENTp. 400 # 11 - 34