4C.FTF.3.4.12.8.11
2011
Domain: Functions
Cluster: Extend the domain of trigonometric functions using the unit circle.
Standards:
3.(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit
circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any
real number.
4.(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Essential Questions
Enduring Understandings
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Why do we create the
unit circle using special
right triangles?
Why do we use the unit
circle?
How can symmetry be
used to explain
mathematical concepts?
Content Statements
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Students will solve
problems involving the
trigonometric functions
of real numbers and the
properties of the sine
and cosine as periodic
functions.
Students will generate
the graphs of sine and
cosine functions.
Students will use the
unit circle to solve
trigonometric functions.
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Mathematical equations can
be solved using inverse
trigonometric functions.
There is a relationship
between sine and cosine and
the unit circle.
Symmetry and periodicity
can be explained using the
unit circle.
Activities, Investigation, and Student Experiences
Use the unit circle to find each value.
1. tan 180°
2. sec (-90°)
3. csc 30°
4. cot (-180°)
5. sin 225°
6. cos 300°
7. tan π/2
8. cot 5π/4
9. sin 7π/4
10. csc 7π/6
 Find the amplitude, period, and phase shift of each
function. Graph each function, showing at least one
period.
a. y = 4sin(2x – π)
b. y =2cos(2πx -4)
4C.FTF.3.4.12.8.11
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Students will
differentiate between
odd and even functions.
Assessments
Example problems:
Find the period of each function.
y = -2 sin x
y = 4 cos (x/3)
y = 1.5 cos 4x
y = -2/3 sin (x/2)
Example open-ended problem.
The function P = 100 – 20cos(5πt)/3 approximates the blood
pressure P in millimeters of mercury at time t in seconds for a
person at rest.
a) Find the period of the function.
b) Find the number of heart beats per minute.
Example graphing problem.
Sketch the graph of the function (include two full periods).
a) y = -2 sin 6x
b) y = -3 cos 4x
c. y = -3sin(2x + π/2)
2011
4C.FTF.3.4.12.8.11
Equipment Needed:
Calculator (graphing)
Smart board
White board
Overhead
2011
Teacher Resources:
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http://www.mathsisfun.com/geometry/unit-circle.html
http://mac.iupui.edu/unitcircle154.pdf
http://www.youtube.com/watch?v=ao4EJzNWmK8
http://mathworld.wolfram.com/Trigonometry.html
http://www.nabla.hr/Z_TrigonometricFunctions-A.htm
http://www.analyzemath.com/trigonometry/periods_trig
onometric.html
http://www.analyzemath.com/trigonometry/periods_trig
onometric.html
http://www.teacherschoice.com.au/maths_library/functio
ns/about_trigonometric_functions.htm
http://www.onlinemathlearning.com/graphing-trigfunctions.html
http://www.sosmath.com/CBB/viewtopic.php?t=596
http://www.analyzemath.com/Trigonometry.html