Lesson 14.1: Graphs of the Sine and Cosine Trig Functions

Learning Goals:

1) How do the sine and cosine curves relate to the unit circle?

2) How can we find the domain and range of a sine or cosine curve?

3) What is the amplitude of a sine or cosine curve?

The sine and cosine functions can be easily graphed by considering their values at the quadrantal angles, those that are integer multiples of or radians. Due to considerations from physics and calculus, most trigonometric graphing is done with the input angle in units of radians, not degrees.

Graphing the sine function: By using the unit circle, fill out the table below for selected quadrantal angles.

Radians

is an odd function because it is reflected over the origin!

Graphing the cosine function: By using the unit circle, fill out the table below for selected quadrantal angles.

Radians

is an even function because it is reflected over the-axis!

The domain and range of the sine and cosine functions are the same. State them below in interval notation:

Domain:

Range:

Critical values for the sine curve:

Critical values for the cosine curve:

Now we would like to explore the effect of changing the coefficient of the trigonometric function. In essence we would like to look at the graphs of functions of the forms:

The grid below shows the graph of . Use your graphing calculator to sketch and label each of the following equations. Be sure your calculator is in RADIAN MODE.

The basic sine function is graphed below. Without the use of your calculator, sketch each of the following sine curves on the axes below.

Amplitude: height of the wave from -axis! Cannot be negative, so amp

1. Write the equation of the following graphs:

a. b.

2. Graph

in the interval

Key Points:

3. Graph one cycle of the function

Key Points:

one cycle

Homework 14.1: Graphs of the Sine and Cosine Trig Functions

1. On the grid below, sketch the graphs of each of the following equations based on the basic sine function.

2. On the grid below, sketch the graphs of each of the following equations based on the basic cosine function.

3. Which of the following represents the range of the trigonometric function

?(1) (2) (3) (4)

4. Which of the following equations describes the graph shown below?

(1)

(2)

(3)

(4)

5. Which of the following equations represents the periodic curve shown below?

(1)

(2)

(3)

(4)

6. Which of the following lines when drawn would not intersect the graph of

?(1) (2) (3) (4)

Graphing Tangent

Learning Goals:

1) How does the tangent curve relate to the unit circle?

2) How do we graph tangent curves with changes to its properties?

1. On your plot, sketch the graph of by using the following table below.

By using the unit circle, fill out the table below for selected quadrantal angles.

Radians

Think about the values of and in order to help you graph

· One cycle occurs between and radians. (asymptotes)

· The length of one cycle (period) is radians.

· There are vertical asymptotes at each end of the cycle. The asymptotes occurs at radians and repeats every radians.

· The range is .

· There is no amplitude.

· The graph is symmetric about the origin, meaning it is an odd function.

How do we graph tangent curves with changes to its properties?

Each set of axes below shows the graph of . Use what you know about function transformations to sketch a graph of for each function on the interval

2. a. b.

c. How does changing the parameter affect the graph of ? Vertical dilation so multiply the -values by

3. a. b.

c. How does changing the parameter affect the graph of ? Horizontal translation (move graph to the left or right) and it is opposite from what you see!

4. a. b.

c. How does changing the parameter affect the graph of ? Vertical translation!

5. a. b.

c. How does changing the parameter affect the graph of ?

Horizontal dilation so you multiply the -value by

Graphing Tangent

1. Each set of axes below shows the graph of . Use what you know about function transformations to sketch a graph of for each function on the interval ).

a. b.

c.

Lesson 14.2: Graphs of the Sine and Cosine Trig Functions –

Period and Frequency

Learning Goals:

1) How do we find the period of a sine or cosine curve?

2) How do we find the frequency of a sine or cosine curve?

3) How do we write the equation of a sine or cosine curve given the graph?

In this lesson, we will explore graphs of the form:

Below is the graph of one cycle of

Below are the graphs of .

· As the value of changes, what happens to the sine curve? Graph gets stretched or compressed horizontally!

· Between , how many cycles of a sine curve appear for each graph? What do you notice about this number compared to value of ?

Frequency and Period of Sinusoidal Curves

· The parent graphs for sine and cosine have a period of .

· By definition, the period of a sinusoidal curve is the horizontal length of one full cyclemust pass through key points!

· If the value of changes then the number of cycles from changes as well.

· By definition, the frequency of a sinusoidal curve is the number of full cycles from . frequency

Since the sine and cosine functions have period , the functions or complete one period as varies from . In other words, to find the period of a sinusoidal curve, simply divide by the value of

Exercise 1: State the frequency and period of each graph.

(2 cycles from ) ( cycle from )

length of one cycle length of one cycle

Exercise 2: State the amplitude, frequency, and period of sinusoidal curve.

a. b. c.

Exercise 3: Determine the period of

Exercise 4: State the amplitude, frequency, and period of each sinusoidal curve given below.

GRAPHING SINUSOIDAL CURVES WITH CHANGES TO FREQUENCY AND PERIOD

1. Identify the key points for a SINE or COSINE curve.

2. Identify all properties of the trig graph.

Amplitude, frequency, and period

3. Graph one cycle of the wave by using the value of the period and then key points of the curve.

4. Use the given interval/directions to determine how many cycles of the function you need.

Model Problem 1: Graph two cycles of

Model Problem 2: Graph one cycle of

Model Problem 3: On the axes below, graph two cycles of a cosine function with amplitude , period , and passing through the point .

Homework 14.2: Graphs of the Sine and Cosine trig functions – Period and Frequency

1. For each function, indicate the amplitude, frequency, and period.

a. b. c.

2. For each function, indicate the amplitude and period.

3. Which sine function has a period of and an amplitude of ?

(1) (2)

(3) (4)

4. What is the minimum value of in the equation ?

(1) (2) (3) (4)

5. Which statement is NOT true regarding the graph of the equation

?

(1) The amplitude is (2) The range is

(3) The -intercept is . (4) The period is .

6. As angle increases from , the value of will

(1) increase from (2) increase from

(3) decrease from (4) decrease from

7. A radio wave has an amplitude of and a wavelength (period) of meters. On the accompanying grid, using the interval , draw a possible sine curve for this wave that passes through the origin.

8. On the axes below, graph one cycle 9. On the axes below, graph

of a cosine function with amplitude , on the interval

period , and passing through the . State the amplitude,

point .frequency, and period.

Lesson 14.3 & 14.4: Vertical Shifts of Sine and Cosine Curves and Phase Shifts, along with Writing Trig Equations

Learning Goals:

1) What is the vertical shift and midline and how do we determine it?

2) How do we graph sine and cosine functions with a translation?

3) What is a phase shift and midline and how do we determine it?

4) How do we write the equation of a trig function from its graph?

· In this lesson we will explore graphs of the form

.

1. Below are the graphs of

· What did the value of do to the graph of ? What type of transformation is this? Moved up 2 units…it is a vertical shift!

· What would the location of the midline (“new” -axis) be after this transformation?

2. Below are the graphs of

· What did the value of do to the graph of ? What type of transformation is this? Moved down 1 unit…it is a vertical shift!

· What would the location of the midline (“new” -axis) be after this transformation?

Summary

· For curves that have the general form , the value represents the translation of a vertical shift (up or down).

· The value of also represents the midline of the trigonometric function. It is the horizontal line that the sinusoidal curve rises and falls above and below by a distance of , the amplitude.

· If given the graph, the equation of the midline can be found by taking the average of the max and min values of the sinusoidal curve.

Directions: State the range and the equation of the midline for each graph.

midline at midline at

How to graph sinusoidal curves with vertical shifts:

1. Graph the SINE or COSINE curve without the vertical shift.

2. Identify the vertical shift and use this to move all the key points up or down units.

3. Label your final graph.

Model Problem 1: Graph and label the function on the interval

. State the range of the function and the equation of the midline.

Vertical shift up 3

midline at

Model Problem 2: On the axes below, graph one cycle of a cosine function with amplitude , period , midline , and passing through the point . State the range of the cosine function.

Vertical shift down 1

midline at

Directions: State the range of each of the following trigonometric functions.

a. b. c.

Or Or

Or

In this lesson we will explore sinusoidal graphs of the form .

Introduction to Horizontal Shifts (Phase Shifts)

· The graphs of sine and cosine are the same when sine or cosine is shifted left or right. Such a shift is referred to as a horizontal shift.

· How could you shift the sine curve below so that it becomes a cosine curve?

· How could you shift the cosine curve above so that it becomes a sine curve?

Summary of Horizontal Shifts of a Sinusoidal Function

· For curves that have the general form and

, the value represents the translation of a horizontal shift (left or right).

· The graphs of sine and cosine are the same when sine is shifted left by radians. Such a shifting is referred to as a horizontal shift.

shift sine to the left to create cosine

shift cosine to the right to create sine

· In mathematics, a horizontal shift may also be referred to as a phase shift

· Remember that with a horizontal shift you must be careful with identifying which way the graph will be translated (moved left or right)!

Practice 1: Identify the horizontal shift of the sinusoidal functions given below:

a. b.

c. d.

How to Write Trig Equations From Graphs

1. Identify if it is a sine or cosine curve: and

. Sine is on midline and Cosine is above/below midline.

2. If the max/min of the curve is not equidistant from the -axis, try to identify the midline. This value goes in for .

3. From the midline, identify the amplitude by looking for the max/min of the curve. This value goes in for .

· Although the amplitude must be positive, determine if will be positive or negative.

4. Determine the value of by looking at either:

· Period – length of one full cycle

· Frequency – number of full cycles per

Practice 1: Write an equation to represent the trigonometric graphs below.

No vertical shift!

Below the midline

One cycle, so

No shift!

Starts on midline

One cycle , so 2 cycles up to

No shift!

Starts on midline

One half cycle , so

No shift!

Starts above midline

One cycle , so

Practice 2: The periodic graph below can be represented by the trigonometric equation where are real numbers. State the values of , and write an equation for the graph.

Vertical shift!

Find the midline first!

(shift)

Practice 3: Find equations of two different functions that can be represented by the graph shown below – one sine and one cosine – using a horizontal translation for one of the equations.

Homework 14.3 & 14.4: Vertical Shifts of Sine and Cosine Curves and Phase Shifts, along with Writing Trig Equations

1. Graph and label the function on the interval . State the range of the function and the equation of the midline.

2. On the axes below, graph one cycle of a sine function with the amplitude , period , midline , and passing through the point . State the range of the sine function.

3. For the function below, indicate the amplitude, frequency, period, vertical translation, and equation of the midline. Graph the function together with a graph of the cosine function on the same axes. Graph one full period of each function.

4. The following graph can be described using an equation of the form . Determine the values of . Justify your answers.

5. When graphed, the line would not intersect the graph of which of the following functions?

(1) (2)

(3) (4)

6. Which of the following functions has a maximum value of ?

(1) (2)

(3) (4)

7. Which statement is incorrect for the graph of the function

?

(1) The period is . (2) The amplitude is .

(3) The range is . (4) The midline is .

8. Write an equation to represent the trigonometric graphs below.

9. State the amplitude, frequency, period, vertical shift and the horizontal shift of the following function:

10. Write the equation of the graph of translated units up and right units.

11. The periodic graph below can be represented by the trigonometric equation

where are real numbers. State the values of , and write an equation for the graph.

12. A student attaches one end of a rope to a wall at a fixed point feet above the ground, as shown in the accompanying diagram, and moves the other end of the rope up and down, producing a wave described by the equation . The range of the rope’s height above the ground is between and feet. The period of the wave is . State the values of and write an equation that represents this wave.

4.

5.

Lesson 14.5: Sinusoidal Regression (Do with next section!)

Learning Goals:

1) How do we use our calculator to write a sinusoidal regression equation for a given set of data?

2) How can we explain the parameter of a trig function in terms of its real-life situation?

Warm-Up: Answer the following question to prepare for today’s lesson.

A box containing coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial.

Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth.

Location of Sinusoidal Regression in the Calculator

· All regression equations can be found using the graphing calculator. All types of regressions on the calculator are prepared in a similar manner.

· When using sine regression, your calculator must be in RADIAN mode

· Your regression options can be found under STATCALC (scroll for more choices)

· “Iterations” is the number of times the calculator will compute the equation. Any number from can be used, but the larger the number the more accurate the equation will be. The default is set at . We will always be using .

Sinusoidal Regression

1. The chart below shows the average daily high temperature for each month for Saugerties, NY

Write a sine regression equation to model the average daily temperature as a function of the month of the year for Saugerties, NY. Round coefficients to the nearest hundredth.

Using your regression equation, what is the predicted average daily high in September, to the nearest degree?

2. Suppose that the following table represents the average monthly ambient air temperatures, in degrees Fahrenheit, in some subterranean caverns in southeast Australia for each of the twelve months in a year. We wish to model these data with a trigonometric function. (Notice that the seasons are reversed in the Southern Hemisphere, so January is in summer, and July is in winter.)

a) Write a sine regression equation to model the average monthly air temperatures as a function of the month of the year for Australia. Round coefficients to the nearest thousandth.

b) Use your regression equation to predict the average monthly air temperature, to the nearest degree, for January of the following year.

Turn and Talk: Look at the graph and the picture of the Ferris wheel. Discuss the following questions with your partner.

· What does the max on the graph tell you about the wheel? Height of the top cart from the ground

· What does the min on the graph tell you about the wheel? Height of the bottom cart from the ground

· What does the midline on the graph tell you about the wheel? Height of the center of the wheel from the ground

· What does the period of 30 tell you about the wheel? 30 minutes to complete one full cycle on the wheel.

Parameters of Trig Functions in a real-world scenario

· Amplitude: height of the wave from the midline

· Midline: horizontal line at the center of the curve

· Period: How long to complete one cycle

· Frequency: how many full cycle up to

· Maximum/minimum: highest and lowest points from the midline

3. The graph below represents the height of a Ferris wheel for one rotation.

a) Explain how you can identify the radius of the wheel from the graph.

Look at the amplitude!

b) If the center of the wheel is feet above the ground, how high is the passenger car above the ground when it is at the top of the wheel?

4. The height of the saddle of a horse above the base of a carousel can be modeled by the equation , where represents seconds after the ride started.

a) How much time does it take for the horse to complete one cycle of motion and return to its starting height? Find the period!

b) What is the maximum height and the minimum height of the horse’s saddle above the base of the carousel? Midline + amplitude!

and so Maximum & Minimum

Homework 14.5: Sinusoidal Regression

1. The average daily temperature (in degrees Fahrenheit) in Fairbanks, Alaska, is given in the table. Time is measured in months, with representing January 1. Write a trigonometric model that gives as a function of , rounding coefficients to the nearest hundredth.

2. For any given day, the number of degrees that the average temperature is below is called the degree-days for that day. This figure is used to calculate how much is spent on heating. The table below gives the total number of degree-days for each month in in Dubuque, Iowa, with representing January.

a) Find a sinusoidal model for the data, rounding coefficients to the nearest thousandth.

b) Use this model to predict the number of degree-days when , to the nearest day.

Lesson 14.6: Modeling with Trig Functions

Learning Goals:

1) How can we write a trigonometric function that models cyclical behavior?

2) How can we describe the parameters of a trig function within the context of the problem?

Warm-Up: Determine if each of the following functions is a cosine or sine function. Then determine if the leading coefficient is positive or negative.

Parameters of Trig Functions in a Real-World Scenario

· You almost always want to use a cosine function to model a real-world scenario because it is easier to locate the maximum point to start the curve than it is to find the point that lies on the midline.

·

·

·

·

·

·

Recall the equation of a trig function is always in the form:

Example 1: In an amusement park, there is a small Ferris wheel, called a kiddie wheel for toddlers. The points on the circle in the diagram to the right represent the position of the cars on the wheel. The kiddie wheel has four cars, makes one revolution every minute, and has a diameter of feet. The distance from the ground to a car at the lowest point is feet. Assume corresponds to a time when car is closest to the ground.

Makes two cycles and is a wave-like function!

Period minute

Diameter

Min value and Max value

Midline:

Starts at the minimum value.

a) Sketch the height function for car with respect to time as the Ferris wheel rotates for two minutes.

b) Find a formula for a function that models the height of car with respect to time as the kiddie wheel rotates. Starts below the midline cosine!

Example 2: Once in motion, a pendulums’ distance varies sinusoidally from feet to feet away from a wall every seconds.

a) Sketch the pendulum’s distance from the wall over a -minute interval as a function of time . Assume corresponds to a time when the pendulum was furthest from the wall.

Makes two cycles and is a wave-like function!

Period seconds

Min value and Max value

Midline:

Starts at the maximum value.

Graph for seconds

b) Write a sinusoidal function for , the pendulum’s distance from the wall, as a function of time since it was furthest from the wall.

Amp

Example 3: The tides in a particular bay can be modeled using a sinusoidal function. The maximum depth of water is feet, the minimum depth is feet and high-tide is hit every hours. Write a cosine function in the form

, where represents the number of hours since high-tide and represents the depth of water in the bay.

Max and Min

Midline

Example 4: A Ferris wheel is constructed such that a person gets on the wheel at its lowest point, five feet above the ground, and reaches its highest point at feet above the ground. The amount of time it takes to complete one full rotation is equal to minutes. A person’s vertical position, , can be modeled as a function of time in minutes since they boarded, , by the equation

. Sketch a graph of a person’s vertical position for one cycle and then determine the values of . Show the work needed to arrive at your answers.

Homework 14.6: Modeling with Trig Functions

1. The High Roller, a Ferris wheel in Las Vegas, Nevada, opened in March 2014. The foot tall wheel has a diameter of feet. A ride on one of its passenger cars lasts minutes, the time it takes the wheel to complete one full rotation. Riders board the passenger car at the bottom of the wheel. Assume that once the wheel is in motion, it maintains a constant speed for the -minute ride and is rotating in a counterclockwise direction.

a) Sketch a graph of the height of a passenger car on the High Roller as a function of the time the ride began.

b) Write a sinusoidal function that represents the height of a passenger car minutes after the ride begins.

c) Explain how the parameters of your sinusoidal function relate to the situation.

d) If you were on this ride, how high would you be above the ground after minutes?

2: Once in motion, a pendulums’ distance varies sinusoidally from feet to feet away from a wall every seconds.

a) Sketch the pendulum’s distance from the wall over a second interval as a function of time . Assume corresponds to a time when the pendulum was closest to the wall.

b) Write a sinusoidal function for , the pendulum’s distance from the wall, as a function of time since it was closest to the wall.

3. Write an equation for the graph shown below:

Lesson 14.7: Modeling with Trig Functions Day 2

Learning Goal:

1) How can we use trigonometric functions to model cyclical behavior?

Warm-Up:

The following function represents the height of a Ferris wheel at any given time minutes. What information can you gather about the Ferris wheel from the function given?

Sketch a graph if you need a visual.

Amp (Radius )

Midline: (center of wheel is 15 feet above the ground)

(highest point above ground)

(lowest point above ground)

(one cycle in 16 minutes)

Example 1: In the classic novel Don Quixote, the title character famously battles a windmill. In this problem, you will model what happens when Don Quixote battles a windmill, and the windmill wins. Suppose the center of the windmill is feet off the ground, and the sails are feet long. Don Quixote is caught on a tip of one of the sails. The sails are turning at a rate of one counterclockwise rotation every seconds.

a) Model Don Quixote’s height, , above the ground as a function of time, in seconds, since he was closest to the ground.

Cosine starts below the midline!

b) After minute and seconds, Don Quixote fell off a sail and straight down to the ground. How far did he fall? Use the equation you wrote to find the height!

Example 2: The tidal data for New Canal Station is shown in the table below. Write a sinusoidal function to model the data for New Canal Station.

start at the min value!

Example 3: A tsunami is a series of ocean waves that send surges up to feet high onto land. They are caused by underwater earthquakes or explosions. The water level in a tsunami will initially fall below its normal level, then rise an equivalent amount above the normal level, finally returning back to its normal level. Assume the period of this cycle to be minutes.

A tsunami is observed from a coastal pier where the normal depth of the water is feet. Following the cycle pattern described above, the tsunami has an amplitude of feet above the normal water depth. The depth of the water will vary sinusoidally with time.

a) Graph one full cycle of the wave, with time in minutes on the -axis and water height in feet on the -axis.

Period minutes

Amp

Midline at:

Max

Min

b) Write a sine function of the form to model this tsunami.

Graph a sine wave…it goes down first!

c) Predict the depth of the water minutes after the tsunami first reaches the pier, to the nearest tenth of a foot.

Plug in the equation

d) What will be the minimum depth of the water during this cycle, and when will it occur?

Example 4: An athlete was having her blood pressure monitored during a workout. Doctors found that her maximum blood pressure, known as systolic, was and her minimum blood pressure, known as diastolic, was . If each heartbeat cycle takes seconds, then determine a sinusoidal model, in the form , for her blood pressure as a function of time in seconds. Show the calculations that lead to your answer.

Homework 14.7: Modeling with Trig Functions Day 2

1. Write an equation that could represent this function.

2. An athlete was having her blood pressure monitored during a workout. Doctors found that her maximum blood pressure, known as systolic, was and her minimum blood pressure, known as diastolic, was . If each heartbeat cycle takes seconds, then determine a sinusoidal model, in the form

, for her blood pressure as a function of time in seconds. Show the calculations that lead to your answer.

3. Rapidly vibrating objects send pressure waves through the air that are detected by our ears and then interpreted by our brains as sound. Our brains analyze the amplitude and frequency of these pressure waves.

A speaker usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric current through the electromagnet, the paper cone can be made to vibrate. By adjusting the current, the amplitude and frequency of vibrations can be controlled.

The following graph shows the pressure intensity as a function of time , in seconds, of the pressure waves emitted by a speaker set to produce a single pure tone.

a. Does it seem more natural to use a sine or cosine function to fit this graph? Explain.

b. Find the equation of a trigonometric function that fits this graph.

4. Evie is on a swing thinking about trigonometry (no seriously). She realizes that her height above the ground is a periodic function of time that can be modeled using , where represents time in seconds. Which of the following is the range of Evie’s height?

(1) (2) (3) (4)

5. Below is a table that shows the average high temperatures for Harrison, NY for each month of the year. Write a trigonometric equation that could fit this data, rounding coefficients to the nearest thousandth.

6. During one cycle, a sinusoid has a maximum at and a minimum at . What is the period of this sinusoid?

(1) (2) (3) (4) (5)

7. Graph and label the function on the interval . State the range of the function and the equation of the midline.

8. For the function below, indicate the amplitude, frequency, period, vertical translation, and equation of the midline. Graph the function together with a graph of the sine function

on the same axes. Graph at least one full period of the function.

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