Periodic Periodic FunctionsFunctions

Sec. 4.3cSec. 4.3c

Let’s consider…

1Light is refracted (bent) as it passes through glass. In the figure, is the angle of incidence and is the angle of refraction. Theindex of refraction is a constant that satisfies the equation

2

1 2sin sin If and for a certain piece of flint glass, findthe index of refraction.

1 83 2 36

sin83 sin 36 sin83

sin 36

1.689

Periodic Functions y f tA function is periodic if there is a positive number c

such that for all values of t in the domain of f. f t c f t The smallest such number c is called the period of the function.

What are some common periodic functions???What are some common periodic functions???

2

The sine, cosine, and tangent functions!!!The sine, cosine, and tangent functions!!!(what are their periods?)(what are their periods?)

Period of sine and cosine:

Period of tangent:

How do w

e kn

ow thes

e???

How do w

e kn

ow thes

e???

General Information about Periodic Functions … Sin, Cos, and Tan

Periodic Function = cyclical, repeating function

Cycle = one complete pattern

Period = horizontal length of one complete pattern

Amplitude = (max-min)/2

Phase Shift = horizontal translation – what will this do to our periodic functions?

Vertical translation – what would this do to our graphs?

Periodic FunctionsFind each of the following without a calculator.

1.57,801

sin2

Rewrite:

57,800sin

2 2

Note: is just alarge multiple of …

sin 28,9002

2

28,900

sin2

1

Periodic FunctionsFind each of the following without a calculator.

2. cos 288.45 cos 280.45 Rewrite:

Note: and wrap to thesame point on the unit circle, so the cosine of one isthe same as the cosine of the other…

cos 280.45 8 cos 280.45

280.45 8 280.450

Periodic FunctionsFind each of the following without a calculator.

3. tan 99,9994

Note: Since the period of the tangent function israther than , is a large multiple ofthe period of the tangent function…

tan4

2

1

99,999

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t1. For any t, the value of cos(t) liesbetween –1 and 1, inclusive.

The The xx-coordinates on the unit circle-coordinates on the unit circlelie between –1 and 1, and cos(lie between –1 and 1, and cos(tt) is) isalways an always an xx-coordinate on the unit-coordinate on the unitcircle.circle.

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t2. For any t, the value of sin(t) liesbetween –1 and 1, inclusive.

The The yy-coordinates on the unit circle-coordinates on the unit circlelie between –1 and 1, and sin(lie between –1 and 1, and sin(tt) is) isalways a always a yy-coordinate on the unit-coordinate on the unitcircle.circle.

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t3. The values of cos(t) and cos(–t)are always equal to each other(recall that this is the check for aneven function).

The points corresponding to The points corresponding to tt and – and –tton the number line are wrapped to pointson the number line are wrapped to pointsabove and below the above and below the xx-axis with the same-axis with the samexx-coordinates -coordinates cos( cos(tt) and cos(–) and cos(–tt) are equal.) are equal.

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t4. The values of sin(t) and sin(–t)are always opposites of each other(recall that this is the check for anodd function).

The points corresponding to The points corresponding to tt and – and –tton the number line are wrapped to pointson the number line are wrapped to pointsabove and below the above and below the xx-axis with exactly-axis with exactlyopposite opposite yy-coordinates -coordinates sin( sin(tt) and sin(–) and sin(–tt) are opposites.) are opposites.

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t5. The values of sin(t) and sin(t + )are always equal to each other.

This is true for all six trigonometric functions!!!This is true for all six trigonometric functions!!!

2

Since is the distance around theSince is the distance around theunit circle, both unit circle, both tt and and tt + get + getwrapped to the same point.wrapped to the same point.

22

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t6. The values of sin(t) and sin(t + )are always opposites of each other(the same is true of cos(t) andcos(t + )).

The points corresponding to The points corresponding to tt and and tt + +get wrapped to points on either end of aget wrapped to points on either end of adiameter on the unit circle. These points are symmetricdiameter on the unit circle. These points are symmetricwith respect to the origin and therefore have coordinateswith respect to the origin and therefore have coordinates((xx, , yy) and (–) and (–xx, –, –yy). Therefore sin(). Therefore sin(tt) and sin() and sin(tt + ) are + ) areopposites.opposites.

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t7. The values of tan(t) and tan(t + )are always equal to each other(unless they are both undefined).

By our previous observation, tan(By our previous observation, tan(tt))and tan(and tan(tt + ) are ratios of the form + ) are ratios of the form

y

x

y

x

and ,and ,

which are either equal to each other or both undefined.which are either equal to each other or both undefined.

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

t8. The sumalways equals 1.

2 2cos sint t

The sum is always of the formThe sum is always of the form for some (for some (xx, , yy) on the) on the

unit circle. Since the equation ofunit circle. Since the equation ofthe unit circle is ,the unit circle is ,the sum is always 1.the sum is always 1.

2 2 1x y 2 2 1x y

In groups of two or three, explain to each other why each of thefollowing statements are true. Base your explanations on theunit circle. Remember that –t wraps the same distance as t, butin the opposite direction.

t cos ,sinP t t

tAt this point, we can use referenceAt this point, we can use referencetriangles and quadrantal anglestriangles and quadrantal anglesto evaluate trig. functions for allto evaluate trig. functions for allinteger multiples of 30 or 45 .integer multiples of 30 or 45 .This leads us to our 16-point unitThis leads us to our 16-point unitcircle, which you must commitcircle, which you must committo memory!!!to memory!!!