Basic Sine and Cosine Curves

Mrs. Maurer

October 29, 2010

Graphing Activity

Organize yourselves into groups of 2-3Gather the following items:

– 1 sheet of art paper– Uncooked spaghetti – Masking tape – Protractors – Meter sticks – Colored markers or pencils – Twine, rope, or yarn (about 7 ft per group)

Graphing Activity

Begin reading Graphs from the Unit Circle (Radians)* worksheet

Complete the steps as indicated– When drawing your circle, make sure you

have a nice, round circleBe sure to have me check your work at

each checkpoint prior to moving onto the next step. Check points look like this: OK________.

*Graphs from the Unite Circle Activity come from NCTM’s Illuminations website http://illuminations.nctm.org/LessonDetail.aspx?id=L785

Graphing Activity

Once complete and as a group, answer the questions from the Questions from the Unit Circle worksheet

Hang graphs up on back wall

*Questions from the Unite Circle Activity come from NCTM’s Illuminations

website http://illuminations.nctm.org/LessonDetail.aspx?id=L785

Basic Sine Curves

The graph of a sine function is called a sine curve.

As we follow the curve from 0 to 2π, we go through one cycle or one period.

Example:– http://en.wikipedia.org/wiki/File:Sin_drawin

g_process.gifDomain: Set of all real numbersRange: [1, -1]

Basic Cosine Curves

The graph of a cosine function is called a cosine curve.

Example:

– Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Cosine_one_period.svg

Domain: Set of all real numbers Range: [1, -1]

Sine vs. Cosine CurvesSine curve is

symmetric with respect to the ______.– Odd function

Cosine curve is symmetric with respect to the _______.– Even function

Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Sine_cosine_one_period.svg

Origin

y-axis

Key Points for Plotting Curves

Key points per period– x-intercepts– y-Intercepts– Maximum points– Minimum points

Key Points of Sine Curves

Intercept

Intercept

Intercept

Maximum

Minimum

Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Sine_one_period.svg

Key Points of Cosine Curves

Maximum

Intercept Maximum

Intercept

Minimum

Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Cosine_one_period.svg

Amplitude

Represents half the distance between the maximum and minimum values.

When given the equation y = a sin bx and y = a cos bx, the amplitude can be found from the given equation:

Amplitude = a

Period of Sine and Cosine

The period is one full cycle from x = 0 to x = 2π.

Given the equations y = a sin bx and y = a cos bx,

Period = 2 π/b

Real-Life Applications of Sine and Cosine Curves

Respiratory cyclesBlood pressure of mercury for a person

at restFuel consumptionAudio sounds

– Acoustics

Acoustic Sound Wave Demo

When reviewing the activity, keep in mind the following:– Which function, sine or cosine is being

represented?– What, if anything, affects the frequency?– What, if anything, affects the period?– What, if anything, affects the amplitude?

http://illuminations.nctm.org/ActivityDetail.aspx?id=37

Acoustics

Mrs. Maurer’s experience– Development engineer

Worked on Pontiac G6, Chevy Malibu and Equinox, Saturn Aura, and GMC Terrain

– SAE Clean Snowmobile Challenge Exhaust and emission control development Published technical papers Networking Trip to Europe

Goals of the Clean Snowmobile Challenge

Environmentally Friendly Clean and Quiet

Snowmobile Maintain/Improve:

– Fuel Economy– Performance– Cost– Appearance– Handling

Kettering Sled

Handling Event

References

Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Cosine_one_period.svg

Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Sine_one_period.svg

Geek3. Wikipedia. October 29, 2010. http://commons.wikimedia.org/wiki/File:Sine_cosine_one_period.svg

http://en.wikipedia.org/wiki/File:Sin_drawing_process.gif

Maurer, N. 2004. SAE Clean Snowmobile Challenge competition.