Trigonometric functions

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

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

Periodic functions

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

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.

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

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

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

Questions: Determine the period and the amplitude of the functions

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  =

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

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

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

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

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

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

Stretches of periodic functions

Stretches of periodic functions

Translations of periodic functions vertical horizontal