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4.1 Graphs of the Sine & Cosine Functions

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A function f is periodic if f (x + h) = f (x) for every x in domain of f Period of f = smallest positive number h One cycle of graph is completed in each period Ex 1) Verify that the graph represents a periodic function and identify its period. Repeats in cycles periodic! Period = 4

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Graph of y = sin x Period = 2π For all x, sin (–x) = –sin x (odd function) Symmetric wrt origin Domain = R Range = [–1, 1] Zeros occur at multiples of π x y 1 –1

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Graph of y = cos x x y 1 –1 Period = 2π For all x, cos (–x) = cos x (even function) Symmetric wrt y-axis Domain = R Range = [–1, 1] Zeros occur at odd multiples of

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The sine and cosine functions are related to each other. They are called cofunctions. Ex 1) Express each function in terms of its cofunction. a) b)

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We will now take a look at how we can transform the basic sine & cosine curves Use Desmos app & the worksheet to help guide us. Open Desmos. Choose, then Trigonometry, and then All the Trig Functions Tap into box 7 and start deleting until all you are left with is box 2 sin (x) We would like to adjust the window so that the x-axis is showing [–2π, 2π] and the y-axis is [–5, 5] Pinch & spread with 2 fingers to get the window just right

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Now look at WS. A graph from [–2π, 2π] is pictured. We already know about parent graphs & transformations. Write down (and then share) what will happen to y = sin x if you graph #1 (and WHY). Now enter #1: ½ sin (x) in box 3 (to get ½, simply type 1 ÷ 2) (to get sin, under tab and tab) Was your guess right?! Now, lets repeat this process with #2 – 4. Please dont go to back side until we are all ready!!

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On back of WS is y = cos x from [–2π, 2π]. We have just reminded ourselves & practiced graphing transformations using sine as the parent graph. Lets see how quickly (and accurately) you can graph the 4 transformations of y = cos x On Your Mark…. All done! Quickly confirm with Desmos We will do more involved transformations later in the chapter … today just the basics! Get Set….GO!!!

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Homework #401 Pg 187 #1, 2, 3, 7, 11, 14, 19–44 all omit 24, 25

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