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Warm-Up Draw the Unit Circle on a sheet of paper and label ALL the parts. DO NOT LOOK AT YOUR NOTES

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So far, in our study of trigonometric functions we have: defined all of them learned how to evaluate them used them on the unit circle So logically, the next step would be to study the graphs of the functions.

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Unit 4: Graphs and Inverses of Trigonometric Functions MA3A3. Students will investigate and use the graphs of the six trigonometric functions. a. Understand and apply the six basic trigonometric functions as functions of real numbers. b. Determine the characteristics of the graphs of the six basic trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. d. Apply graphs of trigonometric functions in realistic contexts involving periodic phenomena. MA3A8. Students will investigate and use inverse sine, inverse cosine, and inverse tangent functions. a. Find values of the above functions using technology as appropriate. b. Determine characteristics of the above functions and their graphs.

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What’s Your Temperature? Scientists are continually monitoring the average temperatures across the globe to determine if Earth is experiencing Climate Change (Global Warming!). One statistic scientists use to describe the climate of an area is average temperature. The average temperature of a region is the mean of its average high and low temperatures.

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The graph shows the average high (red) and low (blue) temperature in Atlanta from January to December. 1. How would you describe the climate of Atlanta? 2. If you wanted to visit Atlanta, and prefer average highs in the 70’s, when would you go? 3. Estimate the lowest and highest average high temperature. When did these values occur? 4. What is the range of these temperatures? 5. Estimate the lowest and highest average low temperature. When did these values occur? 6. What is the range of these temperatures?

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What’s Your Temperature? A function that repeats itself in regular intervals, or periods, is called periodic. a. If you were to continue the temperature graph, what would you consider its interval, or period, to be? b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods.

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Periodicity is Common in Nature Day/night cycle (rotation of earth) Ocean Tides Pendulums and other swinging movements Ocean waves Birth/marriage/death cycle Menstrual cycle Eating and sleeping cycle Musical rhythm Linguistic rhythm Dribbling, juggling Calendars Fashion cycles, for example, skirt lengths or necktie widths Economic and political cycles, for example, boom and bust economic periods, right-wing and left-wing political tendencies

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Periodic Functions A periodic function is a function whose values repeat at regular intervals. –Sine and Cosine are examples of periodic functions The part of the graph from any point to the point where the graph starts repeating itself is called a cycle. The period is the difference between the horizontal coordinates corresponding to one cycle. –Sine and Cosine functions complete a cycle every 360°. So the period of these functions is 360°.

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Whenever you have to draw a graph of an unfamiliar function, you must do it by point-wise plotting, or calculate and plot enough points to detect a pattern. Then you connect the points with a smooth curve or line. Objective: Discover by point-wise plotting what the graphs of f(x)=cos x and f(x)=sin x look like. Sine and Cosine Functions

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Graphing Sine and Cosine Functions The graph of the sine and cosine functions are made by evaluating each function at the special angles on the unit circle. The input of the function is the angle measure on the unit circle. The output is the value of the sine function for that angle. We can “unwrap” these values from the unit circle and put them on the coordinate plane.unwrap

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Exploration: Parent Sinusoids Sinusoid – a graph of a sine or cosine function “sinus” coming from the same origin as “sine,” and “– oid” being a suffix meaning “like.”

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Graphing Calculator Introduction Make sure that your calculator is in degree mode. Graph y = sin Θ. Trace along the graph. What do you observe?y = sin Θ Repeat for y = cos Θ.

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The Graph of Sine x (angles) y (evaluate for sine) 0o0o 90 o 180 o 270 o 360 o

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The Graph of Cosine x (angles) y (evaluate for cos) 0o0o 90 o 180 o 270 o 360 o

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Important Words Sinusoidal axis - the horizontal line halfway between the local maximum and local minimum Convex – bulging side of the wave Concave – hollowed out side of the wave –Concave up –Concave down Point of inflection - point on a curve at which the sign of the curvature (the concavity) changes.

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The Graph of Tangent x (angles) y (evaluate for tan) 0o0o 90 o 180 o 270 o 360 o

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Co-Trig Functions Each of the co-functions relate to the original graph. Plot the “important points” for the sine function on the cosecant graph and then sketch the sine curve LIGHTLY – not in pen!!!

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The Graph of Cosecant x (angles) y (evaluate for csc) 0o0o 90 o 180 o 270 o 360 o

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The Graph of Secant x (angles) y (evaluate for sec) 0o0o 90 o 180 o 270 o 360 o

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The Graph of Cotangent x (angles) y (evaluate for cot) 0o0o 90 o 180 o 270 o 360 o

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Discontinuous Functions The graphs of tan, cot, sec, and csc functions are discontinuous where the function value would involve division by zero. What happens to the graph when a function is discontinuous?

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Trig Function Characteristics As always, we need to talk about domain, range, max and min, etc. You will fill out the table for each of the 6 trig functions (and finish for HW).

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Period The period of a trig function is how long it takes to complete one cycle. What is the period of sine and cosine? What about cosecant and secant? Tangent and cotangent? The period of the functions tangent and cotangent is only 180° instead of 360°, like the four trigonometric functions.

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Domain and Range When we think about the domain and range, we have to make sure we are considering the entire function and not just the part on the unit circle.

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Maximum and Minimum Local max & min Absolute max & min

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Intercepts x-intercepts y-intercepts

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Points of Inflection point on a curve at which the sign of the curvature (the concavity) changes. Will all graphs have points of inflection?

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Intervals of Increase and Decrease Positive slope? Negative slope?

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PeriodDomainRangeMaximumMinimum Sine Cosine Tangent Cotangent Cosecant Secant

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Increasing Interval Decreasing Interval X-intercept(s)Y-intercept(s)Point of Inflection sine cosine Tangent Cotangen t cosecant secant

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FunctionPeriodDomainRange y = tan Θ180° Θ = all real #’s of degrees except Θ = 90° + 180°n, where n is an integer y = all real #’s y = cot Θ180° Θ = all real #’s of degrees except Θ = 180°n, where n is an integer y = all real #’s y = sec Θ360° Θ = all real #’s of degrees except Θ = 90° + 180°n, where n is an integer y = csc Θ360° Θ = all real #’s of degrees except Θ = 180°n, where n is an integer

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What’s Your Temperature? Sine and Cosine functions can be used to model average temperatures for cities. Based on what you learned about these graphs, why do you think these functions are more appropriate than a cubic function? Or an exponential function?

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Graphs of Other Trigonometric Functions

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