Graphs of Sine and Cosine Functions You’ll need graph paper 4.5

 On Graph paper use a radius of 7in to represent the radius of your unit circle.  Then give both the fractional and decimal value of your trig function for each value of theta on the unit circle. (Do not include the last value of theta for your quadrant) i.e. 90, 180, 270, Unit Circle Activity

 Groups 1-4 – Sine  Groups 5-8 – Cosine  Groups 9-12 – Tangent  Group 4n+1 – Q1  Group 4n+2 – Q2  Group 4n+3 – Q3  Group 4n+4 – Q Groups Ex: Group 7: 7 = 4(1)+3 Cosine Q3 n is a whole # Only need one graph per group

 Label with “as increases, [trigfunction( )] increases/decreases.”  Ex for sine of Q1: “as increases, increases.” 4

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Graph the sine and cosine functions  Graph the sine function on a new piece of graph paper  Label your x-axis in radians in multiples of. Use 1 square for each measure  On your y-axis label. Count 1 square as ¼.  Then graph the cosine function a separate axis. (use the same labels) 6 Everyone needs to do their own! Everyone is starting with sine 13 squares on each side of y-axis 8 squares on each side of x-axis

Graph the csc and sec functions  Graph csc with the sin graph and sec with the cos graph  Then make a new graph for tan and cot Label your x-axis in radians in multiples of 7

8 Ordered Pairs  Consider the values for x and y in the table to the right  Note Period = 2 π Maximum y values Minimum y values xsin(x)cos(x)

9 Graphing the Ordered Pairs Period = 2 π Maximum and minimum values

10 Graphing on Calculator  Go to ♦Y= screen Enter function  Choose F2, zoom 7-Trig  Graph is plotted Tic marks are in units of π/2 Try Web Graphing Utility

11 Amplitude  Defined as the absolute value of maximum or minimum of the function  Try graphing y = 2 sinx What is the amplitude  For y = a cos x or y = a sin x The amplitude is |a| Do we need to worry about the amplitude for the other trig functinos? amplitude = 1 y=sinx 2

12 Period of a Trig Function (Recall slide from previous lesson)  The functions repeat themselves  The period is the smallest value, p for which f(x) = f(x + p)  For sin, cos, sec, csc The period is 2 π  For tan and cot The period is π

13 Period of a Trig Function  What happens for ?  Try graphing y = sin 2x What is the period? What about y = sin 3x  Try y = cos 0.5x What is the period?  For Period = Same for cos, sec, csc

14 Period of a Trig Function  For tangent Note amplitude is without bound Period is π  For Period = Predict the period for y = tan (1/3 x) Graph it and verify your prediction Predict the period for y = tan (1/3 x) Graph it and verify your prediction Same for cot

Review of Transformations 15 Or reflection over x-axis! Or reflection over y-axis!

Review of Transformations 16 Sketch the graph Do non-rigid transformation 1 st (strech/compress) Then rigid transformations (up/down and left/right)

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Let’s investigate with graphs of trig functions! 18 Desmos.com

Practice time  Graph a. b. 19 Sketch each transformation of the graph Sketch between 0 and 2pi while doing transformations Does the make a difference?

20 H Dub  4-5 Pg 328 #1-25odd, 35-51EOO

Graph the sine and cosine functions  Regraph the sine and cosine functions on two separate axis  Label your graph in radians  On your y-axis label Leave room above and below!  On your x-axis label multiples of 21