Section 3.3 Graphing Sine, Cosine, and Tangent Function

SECTION 3.3 GRAPHING SINECOSINE AND TANGENTFUNCTIONS

I) WHAT IS A PERIODIC FUNCTION?

A function that repeats its output values in a cycle

The range of input values within a cycle is known as theperiod

There are many examples of periodic functions in ourworld

Ocean waves and ocean currents

Solar system

Finance and Market trends

Sound waves, light transmission, radiation

Wave propagation: earthquakes,

Heart beat

II) COMPONENTS OF A PERIODIC FUNCTION:

Crest

Trough

Period

Crest: Local maximums on the periodic function. Top

Trough: Local minimums on the periodic function. Bottom

Amplitude: The vertical difference between the top and the middle

Period: The horizontal difference between two rests or two troughs

Note: The horizontal distance between the crest and trough is HALF the

period.

Amplitude

Period

½ of Period

Ex: Given each graph, indicate the period and amplitude

-3

-2

-1

-4

-3

-2

-1

-3

-2

-1

-4

-3

-2

-1

Period

Amplitude

Period

Amplitude

Ex: If high tide occurred at 7:30am and low tide was at1:30pm, what time will the next high tide be?

The time difference between high tide and low tide is half the period

The time for a full period will be 12 hours

So the next high tide will be 12 hours from 7:30am

HIGH AND LOW TIDE:

At high tide, the ocean levels go up

At low tide, the ocean levels go down

Ocean levels will fluctuate

up and down over several hours

III) SINE FUNCTION IN AN UNIT CIRCLES:

Sinθ in an unit circle is equal to the y-coordinate of point “P”

The highest y-coordinate is at 1 (at 90°) & the lowest at -1 (at 270°)

The value of Sinθ changes periodically from 1 to -1 depending onthe value of the angle

If we graph the sine function, we are comparing how height of pointP changes in relation to the angle in standard position

Y-coordinate

Note: hyp= 1

IV) GRAPHING THE SINE FUNCTION:

X-axis: value of the central angle in radians

Y-axis: value of sinθ, height of point “P”

Highest (1) and lowest (-1)

-0.5

0.5











-1

SINE FUNCTION:











-1

-0.5

0.5

As you rotate the terminal arm, the value of the angleincreases

The height of the graph changes fluctuates up anddown in accordance to the angle

The result is a wave function

Ex: Find the following for the sine function:

a)Period & amplitude

b)Y intercept

c)X intercept and a general formula

d)Domain and Range

-0.5

0.5











Period

Period:

Amplitude

X-intercepts:

Y-intercepts:

Domain:

Range:

V) COSINE FUNCTION IN AN UNIT CIRCLES:

Cosθ in an unit circle is equal to the x-coordinate of point “P”

The lowest x-coordinate is -1 (at 180°) & the highest is 1 (at 360°)

The value of Cosθ changes periodically from 1 to -1 depending onthe value of the angle

If we graph the Cosine function, we are comparing the horizontaldistance of point P vs the angle in standard position

x-coordinate

Note: hyp= 1

VI) GRAPHING THE COSINE FUNCTION:

X-axis: value of the central angle in radians

Y-axis: value of cosθ, x-coordinate point “P”

Highest (1) and lowest (-1)

-0.5

0.5











-1

COSINE FUNCTION:











-1

-0.5

0.5

Rotate your unit circle, so you can measure the horizontaldistance of point P

The Cosine graph will then begin at the top

Then you graph how the horizontal distance of P changesin relation to the angle in standard position

The result is a wave function

Ex: Find the following for the cosine function:

a)Period & amplitude

b)Y intercept

c)X intercept and a general formula

d)Domain and Range

-0.5

0.5











Period

Period:

Amplitude

X-intercepts:

Y-intercepts:

Domain:

Range:

VII) SUMMARY FOR GRAPHING COSINE/SINE FUNCTIONS

When graphing a Sine & Cosine function, there are only 5points to consider: Beginning, Middle, End, Quarter way,and 3 Quarters way of the period

Sine Function:

At the Beginning, Middle, and End of the period, thewave will cross the x-axis

At quarter way, the graph will be at the top

At 3-quarters way, the graph will be the bottom

Cosine Function

At the Beginning and End of the period, the graph will beat the top

In the Middle, the wave will be at the bottom

At quarter way & 3 quarters way, the graph will be on thex-axis

Ex: What is the equation for the following graph?

-0.5

0.5











The graph is a sine function with avertical reflection

REVIEW: TANGENT FUNCTION & TANGENT LINES

SOH-CAH-TOA

Using the ratio of opposite over adjacent, we can showthat the tangent function is also sine over cosine

A tangent line is a line that crosses a circle at onlyone point

Tangent Line

Point of Tangency

VIII) TANGENT FUNCTIONS AND UNIT CIRCLES

There are two ways to relate the tangent function with anunit circle

1st Method

The tangent function can be defined as the y-coordinate ofPoint P, divided by its x-coordinate

2nd Method

The tangent function can be defined as the y-coordinate ofthe extension of the terminal arm on a vertical tangent line

IX) TANGENT = Y-COORDINATE / X-COORDINATE

The tangent function is the ratio of the Y-coordinateover the X-coordinate

As we rotate the around the unit circle, the ratiochanges according to the angle in standard position

X) TANGENT AS THE Y-COORDINATE ON THE VERTICALTANGENT LINE

The tangent function can be defined as the y-coordinate of theextension of the terminal arm on a vertical tangent line

When you rotate the terminal arm,the extension creates a righttriangle with a base of 1

The tangent function will be theY-coordinate of this point on theVertical tangent

As the terminal arm rotates around the circle, it will cross thetangent line above the X-axis (+’ve) or below the X-axis (–’ve )

Ie: when the terminal arm is in Q1, it crosses the tangent lineabove the X-axis, so tanθ is positive

In Q2, the terminal armextends to the tangent linebeneath the X-axis, so itwill be negative

In Q3, it crosses thetangent line above the X-axis, so tanθ is positive

In Q4, it crosses thetangent line under the X-axis, so tan θ is negative

At any point when theterminal arm is pointingside ways, the value of tanθis equal to zero

At any point when it’spointing up or down, thevalue of tanθ is undefined,because it doesn’t touch thetangent line

XI) TANGENT FUNCTION AS A RECIPROCAL

The Tangent function is like a reciprocal function

At points where cosθ = 0, you will get a vertical asymptotebecause the reciprocal of zero  Undefined

At points where sinθ = 0, you will be a x-intercept

At points where sinθ = cosθ , the value of tanθ will beequal to 1 (ie: 45°, 135°, 225°, and 315°, reference angleequals 45°)









-3

-2

-1

TANGENT FUNCTION





























-2

The tangent function can be defined as the y-coordinate ofthe extension of the terminal arm on a vertical tangent line

As the angle changes and increase, the value of tanθ willfluctuate from negative to positive infinity

Ex: Indicate the following for the tangent function:

a)Domain & Range

b)X-intercepts and the general formula

c)Period and amplitude









-3

-2

-1









XII) VERTICAL ASYMPTOTE AND GENERAL FORMULA:

Since tanθ = (sinθ/cosθ), the vertical asymptotes will allappear when cosθ=0 (Denominator is equal to zero)

The general formula for all the vertical asymptotes in atangent function will be:

Ex: Given that 0< θ< 2π , tanθ > 0 and sinθ < 0, whatquadrant is the angle in?

If tanθ > 0 (positive), then it can only be in quadrants 1 and 3

If sinθ < 0 (negative), then it can only be in quadrants 2 and 3

To satisfy both conditions, then the angle must be in quadrant 3

HOMEWORK:

Assignment 3.3