Download - Nmaurer sine cosine lesson
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.