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TRIGONOMETRIC PARENT FUNCTIONS AND THEIR GRAPHS Sections 4.5 and 4.6 (selected)

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THINK?????? Give an example of a behavior that repeats a particular pattern over and over.

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Seasons & Daylight Hours

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High & Low Tides Heartbeats Sleep Patterns Motion of a Spring as it goes to equilibrium Yearly Temperatures at a particular location Swing of a Pendulum http://mw.concord.org/modeler1.3/mirror/mechanics/pendulum.gif http://mw.concord.org/modeler1.3/mirror/mechanics/pendulum.gif

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“Periodic” Behavior that repeats a pattern infinitely Period is the time it takes to complete one full cycle Example: The ocean level at a beach varies from low tide to high tide and then back to low tide approximately every 12 hours. Trigonometric functions are used to model periodic phenomena. Why? By starting at point P(x, y) on the unit circle and traveling a distance of 2 units, 4 units, 6 units, and so on, we return to the starting point P(x, y). Because the trigonometric functions are defined in terms of the coordinates of that P, if we add (or subtract) multiples of 2 to t, the values of the trigonometric functions of t do no change.

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http://www.xpmath.com/careers/topic sresult.php?subjectID=4&topicID=14 http://www.xpmath.com/careers/topic sresult.php?subjectID=4&topicID=14 Careers that use trigonometric functions

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RECALL In a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc, both of which are given by the same real number t. For each real number t, there corresponds a point P(x, y) on the unit circle. In terms of a function, the input is the real number t and the output involves the point P(x, y) on the unit circle that corresponds to t.

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INTRODUCTION Trigonometric Functions (aka Circular Functions) can be graphed in a rectangular coordinate system by plotting points whose coordinates satisfy the function We will use the traditional symbol, x, to represent the independent variable, measured in radians and y or f(x) to represent the dependent variable http://demonstrations.wolfram.com/SineCos ineTangentAndTheUnitCircle/ http://demonstrations.wolfram.com/SineCos ineTangentAndTheUnitCircle/

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Characteristics of y = sin(x)

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Real-World Applications Sound Waves (Music, Radio, TV, etc.)

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Real-World Applications Yearly Temperature at Central Park (or any specified location

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Characteristics of y = cos x

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Real World Applications Brain Waves

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Real World Applications Tides

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Characteristics of y = tan x

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Real World Applications http://catcode.com/trig/trig13.html

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Characteristics of y = csc x Function, passes vertical line test Domain: D –all real numbers except integral multiples of π Range: R (-∞,-1] U [1, ∞) Is periodic (pattern repeats). The period is 2π Is odd (symmetric to origin ) No Zeros No Y-intercept Vertical asymptotes occur at integral multiples of π

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Characteristics of y = sec x

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Characteristics of y = cot x

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http://www.youtube.com/watch?v=RmYv0xsz31k

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Graphs of Other Trigonometric Functions 4.6

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