1. Trigonometric functions

2. the smallest value of p is the period of the function
Periodic behaviour
Any function is called periodic if it “repeats” itself on intervals of any fixed length. For example the sine curve.
Periodicity may be defined symbolically:
A function f is periodic if there is a positive number p such that
f (x+p) = f(x) for every x in the domain of f.
the smallest value of p is the period of the function

3. Periodic behaviour in nature
wave motion: light, sound
tides: cyclic rise and fall of seawater
water waves
seismic tremors

4. Periodic functions

5. Periodic motion
Motion that repeats itself over and over is called Periodic Motion or Oscillation.
It always has a stable equilibrium position
1 complete revolution
1period
1period

6. Periodic Function
Amplitude is the maximum multitude of displacement from equilibrium. It is always positive.
A = max value - min value 2
Period is the the time to complete one cycle.
Cycle is one complete round trip from A to -A then back to A.

7. Periodic Function
Axis of the curve is the horizontal line that is half way between the maximum and minimum values of the periodic curve.
y = maximum value + minimum value 2

8. Periodic behaviour in physics
motion of a spring and a block
common back-and-forth motion of a pendulum
bouncing ball
circular motion

9. Periodic behaviour in life
radio waves
clock mechanism
repeated steps of a dancer
ballet Don Quixote (32 fouette turns)
music

10. Questions:
Determine the period and the amplitude of the functions

11. Trig ratios of any angle
( x 2 + y 2)
sin  = y
P (x, y) x
( x 2 + y 2)
cos  =  y x x y x
tan  =

12. Trig ratios of any angle
(x 2 + y 2)
sin  =
positive y x
( x 2 + y 2)
P (x, y)
cos  =
negative y  y x
negative
tan  = x x
all ratios are positive
only sin is positive
II S
A I
only tan is positive
III T
C IV
only cos is positive

13. Special angles Special angles: 30o, 60o, 45o, 90o.
Two special triangles can be used to find the exact values of the sine, cos, tan of special angles.
45o
30o
 2 2 1 3
45o
60o 1 1

14. Radians and angle measure
1 radian is the measure of the angle subtended at the centre of a circle by arc equal to the radius of the circle r
 = 1 radian 
r = a / 
a =  * r r
 = a / r
angle in radians

15. Graph of f(x) = sin x 1 Domain : all real numbers, R.
2 Range :  y  1 .

16. Graph of f(x) = cos x 1 Domain : all real numbers, R.
2 Range :  y  1 .

17. Graph of f(x) = tan x
1 Domain : all real numbers, R, x  /2  odd number.
2 Range : all real numbers, R .

18. Stretches of periodic functions

19. Stretches of periodic functions

20. Translations of periodic functions
vertical
horizontal