The World’s Largest Ferris Wheel

• In London, British Airways built a ferris wheel which measures 450 feet in diameter. It turns continuously at a constant rate, completing a single rotation every 30 minutes.

• The values of f(t), your height above the ground t minutes after boarding the wheel, are shown in the table below:

• The graph of f(t) is shown on the next slide.

t, min 0 7.5 15 22.5 30 37.5 45 52.5 60

f(t), feet 0 225 450 225 0 225 450 225 0

A function f is periodic if f(t+c) = f(t) for all t in the domain of f . The smallest positive constant c for which this relationship holds is called the period of f. A part of the graph of f as t varies over an interval of one period length is called a cycle.

The midline of a periodic function is the horizontal line midway between the function’s maximum and minimum values. The amplitude is the vertical distance between the function’s maximum (or minimum) and the midline.

Period = 30

Midline: y = 225Amplitude = 225

The Unit Circle

• The unit circle is the circle of radius one that is centered at the origin. The equation of the unit circle is x2 + y2 = 1.

• Angles (denoted by in the figure below) can be used to locate points on the unit circle. These angles are measured counterclockwise from the positive x-axis.

(1,0)(-1,0)

(0,-1)

(0,1)

P = (x,y)

θ

θ

x

y

Definition of Sine and Cosine

• Suppose P = (x,y) is the point on the unit circle specified by the angle . We define the functions , cosine of , or , and the sine of , or , by the formulas

See figure below.

θ cosθ θ

θ θsin

y. θsin and x cos θ

x

y

y)(x, P

θ

1

xθ cos

y θsin

Values of the Sine and Cosine Functions

• Find the values of:

• In each case, find the point on the unit circle determined by the given angle. The coordinates of this point are the values we seek.

• We have that Similarly, However, in order

to evaluate we must use a calculator. We find that

.210sin ,210 cos ,180sin ,180 cos ,90sin ,90cos

1. 90sin and 0 90cos 0. 180sin and 1 180cos

,210sin and 210cos . 5.0210sin and8660 210cos .

Coordinates of a point on a circle of radius r.

• The coordinates (x,y) of the point Q in the figure below are given by:

.θsin r y and θ cosr x

θ

)θsin ,θ (cos

)y ,x( Q

1

r

Height on the ferris wheel as a function of angle

• The ferris wheel described earlier has a radius of 225 feet. Find your height above the ground as a function of the angle measured from the 3 o’clock position.θ

P = (x, y)

θ θ225sin 225 Height

θ225sin

y

x

225

Radians

• We have previously measured angles in degrees, but now we introduce a new way to measure angles--in radians.

• The arc length spanned, or cut off, by an angle is shown next:

• An angle of 1 radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length 1. An angle of 1 radian is approximately equal to

θby spannedlength Arc θ

x

y

.57.3

• The radian measure of a positive angle is the length of arc spanned by the angle in a unit circle. For a negative angle, the radian measure is the negative of the arc length.

• To convert from degrees to radians,

multiply the degree measure by

• To convert from radians to degrees,

multiply the radian measure by

• Examples.

• Problem. Convert 4 radians to degrees. In which quadrant is this angle?

.180

radians. 0.01745 1 and 90- radians /2-

.180

Arc Length

• The arc length, s, spanned in a circle of radius r by an angle of radians, is given by

• Problem. What length of arc is cut off by an angle of 120 degrees on a circle of radius 12 cm? Solution. First convert to radians, then multiply by 12 to get an arc length of

θ ,2 θ 0 .rs θ

radiansθ

θr length Arc r

x

y

cm. 8

Effect of tire wear on mileage

• The odometer in your car measures the mileage travelled.

• The odometer uses the angle that the axle turns to compute the mileage s in the formula

• Question. With worn tires, are the actual miles travelled more or less than the odometer miles?

θ .rs θ

Values of Sine and Cosine

• The values for sine and cosine in the following table of “standard” angles (see pg 301 of text) should be memorized:

• You should be able to use reference angles (see next slide) to find the values of sine and cosine for angles which are not in the first quadrant. For example,

/2 /3 /4 /6 0 radians θ 1 2/3 2/2 1/2 0 θsin

0 2/1 2/2 /23 1 θ cos

.2/3 /65 cos and 1/2 /65sin

Reference Angles

• For an angle corresponding to the point P on the unit circle, the reference angle of is the angle between the line joining P to the origin and the nearest part of the x-axis. A reference angle is always between that is, between (See page 291 of the textbook.)

• Example. Find the reference angle of

θ θ

,90 and 0 /2. and 0

.6/5

/65 θ )2/1 ,2/3( P

/6 angle reference x

y

The Unit Circle

)1,0(),( 2

321

),( 21

23

),( 22

22

),( 23

21

),( 22

22

),( 21

23

),( 21

23

),( 22

22

),( 23

21

),( 21

23

),( 22

22

),( 23

21

)0,1(

)0,1(

)1,0(

90 2

180

23 270

2 3600 0

30

45

60

6

4

3 3

2

43

65

120

135

150

611

47

35

67

45

34

330

315

300240

225

210

x

y

For any angle :

(x,y) = (cos , sin )

Graphs of sin x and cos x, where x is in radians

• The function sin x is periodic with period

• The function cos x is periodic with period

.2

.2

Red arc length is π/3.

Generation of Sine Curve Using Unit Circle

Graph of sin x, where x is in radians

• One cycle of the function sin x.

Graph of cos x, where x is in radians

• One cycle of the function cos x.

Graph of ferris wheel function

• The graph of the function giving your height, y = f(x), in feet, as a function of the angle x, measured in radians from the 3 o’clock position, is shown next.

ft. 225 Amplitude

Period = 2π

Midline y = 225 ft.

f(x)=225+225sin x

Sinusoidal Functions• We consider functions which can be expressed in the form:

where A, B, h, and k are constants and t is measured in radians. The graphs of these functions resemble those of sine and cosine. They may start with sine or cosine and then shift, flip, or stretch the graph. Which of these transformations are to be applied is determined by the values of the constants A, B, h, and k as follows: • |A| is the amplitude

• • h is the horizontal shift • y = k is the midline • |B| is the number of

cycles completed in •

k,h)B(t cosA y andk h)B(tsin A y

period theis |B| /2

.2t 0 frequency -the-unit timeper cycles ofnumber is /2|B|

One Cycle of

• Assume A, B, h, and k are positive.

kh)-B(tsinAy

B

2

h

Ak

Ak

k

hB

h

B

2h

t

y

A

The ferris wheel function as a sinusoidal function

• Problem. Use the sinusoidal function to represent your height f(t) above ground while riding the ferris wheel. Solution. The diameter of the ferris wheel is 450 feet so the midline is k = 225 and the amplitude, A, is also 225. The period of the ferris wheel is 30 minutes, so

The graph is shifted to the right by t = 7.5 minutes since we reach y = 225 (the 3 o’clock position) when t = 7.5. Thus, the horizontal shift is h = 7.5, and the formula for f(t) is:

where t is in minutes and height is in feet.

.1530

2 B

225, 7.5)t (15

sin 225 f(t)

Phase shift

• Assume that a sinusoidal function has been given. The phase shift is defined as Then since

it follows that the ratio of the phase shift to equals the fraction of a period by which the sinusoidal function is shifted.

• For sinusoidal functions written in the form

2

,22

hB

/B)(2

h

),cos(BtAy and )sin(BtAy

shift. phase theis

hB.

Phase shift, continued• Phase shift is significant because we often want to know if

two waves reinforce or cancel each other. For two waves of the same period, a phase shift of 0 or 2 tells us that the two waves reinforce each other while a phase shift of tells us that the two waves cancel.

• Example. The graph of is the same as the graph of but shifted /12 units to the left, which is ( /4)/(2) = 1/8 of the period of cos(3t).

/4)(3t cos (3t) cos

cos(3t)cos(3t +/4)

Sine with a phase shift of π/2

• plot({sin(x),sin(x+Pi/2)},x=-Pi..Pi,color=black);

• Do you recognize the shifted sine curve?

The Tangent Function• Suppose P = (x, y) is the point on the unit circle specified

by the angle . We define the tangent of , or tan , by tan = y/x. The graphical interpretation of tan is as a slope. In the figure below, the slope m of the line passing from the origin through P is given by

θ θ θ θ

.tan m θ

θ. tan x

y slope has Line

y) (x, P

θ

θ

y

x 0) (1,

1) (0,

Values and graph of the tangent function• The values of the tangent function for “standard” angles are:

• The function tan x is periodic with period , and its graph is next. It has a vertical asymptote when x is an odd multiple of

/2 /3 /4 /6 0 radians x undef 3 1 31/ 0 xtan

.2/

More trigonometric functions

• In addition to the tangent function, we define the following trigonometric functions:

• Since these new functions are defined as reciprocals of previously studied functions, their graphs can be obtained by transforming the graphs of the previously studied functions.

θ/sin cos θ1/tan θcot θcotangent

θ1/sin θ csc θcosecant

θ 1/cos θ sec θsecant

θ

Relationships among the trigonometric functions

• A number of so-called trigonometric identities are known for the trigonometric functions we have defined. These are equations expressing a relation among trig functions which holds for all values of for which the equations are defined.

• Examples of trigonometric identities are:

θcsc 1 θcot

θsec θ tan 1

1θsin θcos

θ θ/cossin tanθ

22

22

22

θ

Inverse Trigonometric Functions

• Solving equations involving trigonometric functions can often be reduced (after some algebraic manipulation) to an equation similar to the following example:

Find a value of t radians satisfying cos t = 0.4 .

• In order to find a value of t as in the above example, we can use a calculator which has support for inverse cosine. Such a calculator in radian mode allows us to find the required value of t by pressing the button labeled as shown below:

• Pressing this button yields cos-1(0.4) 1.16, where cos(1.16) 0.4 .

cos-1

The Inverse Cosine Function.

• The inverse cosine function, also called the arccosine function, is denoted by cos-1y or arccos y. We define

In other words, if t = arccos y, then t is the angle between 0 and whose cosine is y. The inverse cosine has domain

• In conjunction with the inverse cosine, we may have to use reference angles to find answers which are not in

• Example. cos-1(0.4) 1.16 gives us an answer in the first quadrant. Suppose we want an answer in the fourth quadrant. We simply subtract the reference angle from

to obtain t 5.12 .

. t 0 and t cos y that providedy cos t -1

. t 0 range and 1 y 1

. t 0

2

The Inverse Sine Function.

• The inverse sine function, also called the arcsine function, is denoted by sin-1y or arcsin y. We define

The inverse sine has domain

• In conjunction with the inverse sine, we may have to use reference angles to find answers which are not in its range.

• Example. sin-1(0.707) = gives us an answer in the first quadrant. Suppose we want an answer in the second quadrant. We simply subtract the reference angle from

to obtain t = .

.2/ t 2/ andsin t y that providedy sin t -1

4/

range and 1 y 1

.2/ t 2/

4/3

The Inverse Tangent Function.

• The inverse tangent function, also called the arctangent function, is denoted by tan-1y or arctan y. We define

The inverse tangent has domain

• In conjunction with the inverse tangent, we may have to use reference angles to find answers which are not in its range.

• Example. tan-1(1.732) = gives us an answer in the first quadrant. Suppose we want an answer in the third quadrant. We simply add the reference angle to to obtain t =

.2/ t 2/ and tan t y that providedy tant -1

3/

range and y

.2/ t 2/

3./4

• While riding the ferris wheel, at which times during the first turn is your height = 400 feet? As shown in the text, we must find two solutions for t between 0 and 30 in the equation

The first solution is given by the arcsin. We solve for t in the equation

This yields t = 11.75 minutes. The second solution corresponds to another angle on the circle with the same reference angle, 0.891, and a positive value of the sine. This is in the second quadrant, so we have

.225

175)5.7(t

15sin

. 0.891 225

175arcsin

2t

15

min. 18.25 t yields which , 0.891 2

t15

Summary for Trigonometric Functions• Periodic function, period, midline, and amplitude were defined.

• If (x,y) is a point on a circle of radius r specified by angle , then

• Radian measure of an angle was defined and

• The arc length, s, spanned by an angle of radians in a circle of radius r is

• Values of sine and cosine for “standard” angles were given.

• Reference angles can be used to find values of sine and cosine for angles which are not in the first quadrant.

• The sine and cosine functions are periodic with period and their graphs are similar but shifted horizontally.

• For sinusoidal functions, we related amplitude, period, hor. shift, and midline to the parameters of the function.

• Phase shift can be used to determine the fraction of a full period by which a sinusoidal graph is shifted.

θ .θsin r y and θ cosr x

.180 radians

θ.r s θ

2

Summary for Trigonometric Functions, cont’d.

• For a point (x, y) on the unit circle specified by angle , tan was defined as y/x.

• The values of tangent for standard angles were given.

• The function tan x is periodic with period and it has vertical asymptotes when x is an odd multiple of

• Three other trig functions were defined as reciprocals: secant, cosecant, and cotangent.

• Four trig identities were given.

• The inverse functions for cosine, sine, and tangent were defined and their domains and ranges were given.

• Use of the inverse trig functions in solving problems was illustrated.

θ θ

.2/