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# FUNCTNS TRIG & LNR STM MATH 150

Texas A&M

GPA 3.6

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This 4 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 150 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/226034/math-150-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 150 Section 53 Lecture Notes Graphs of the Sine and Cosine Functions Recall that the circumference of the unit circle is 27 This means that the terminal point Px y determined by t is the same as that determined by t 27E Since the sine and cosine functions are de ned in terms of the coordinates of Px y it follows that sint 27m sint for any integer n cost 27m sint for any integer n This means that the sine and cosine functions are periodic De nition A function f is periodic if there is a positive number p such that f t p f t for every t The smallest such positive number is called the period of f De nition If a function f has period p then the graph of f on any interval of length p is called one complete period of f Periodic Properties of Sine and Cosine The function sine has period 27E sint 27c sint The function cosine has period 27 cost 27c cost Sketching the Graphs of Sine and Cosine Recall that by de nition sint y where y is the ycoordinate of the terminal point on the unit circle deter mined by t and cost x de ned similarly t sint cost 0 gt 0 gtl l gt0 g M E l gt0 0 gt l na37 0a 1 1a0 37 gt27c 1a0 0A1 Graphs of y sint and y cost with Multiple Periods AANMA VV VVV A AAHAA VVVVVV NOTE The graph of y sint is symmetric with respect to the origin since sine is an odd function The graph of y cost is symmetric about the yaXis since cosine is an even function Graphs 0f Transformations of Sine and Cosine De nition For functions of the form f t asint and f t acos t the amplitude A of the function is the largest value of y ieA a In general A max value of ft min value 0fft Example 1 Sketch the graph of ay 2 sinx and b y sinx 2 Example 2 Sketch the graph of y 3 cosx Answer Example 3 Sketch the graph of y sinx Sine and Cosine Curves The sine and cosine curves y asinkx andy acoskx k gt 0 27c have amplltude a and perlod An approprlate 1nterval on wh1ch to graph one complete perlod 1s 0 Example 4 Find the amplitude and period fo y sing and graph one complete period Example 5 Find the amplitude and period fo y 2cos 2x and graph one complete period NOTE The graphs of functions of the form y asinkx b and y acos kx b are simply sine and cosine curves shifted horizontally by an amount of b They are shifted to the right if b gt 0 or to the left if b lt 0 De nition The number b is called the phase shift Shifted Sine and Cosine Curves The sine and cosine curves y asinkx b andy acoskx b k gt 0 27 have amplitude a period and phase shift b An appropriate interval on which to graph one complete 2 period is b b Example 6 Find the amplitude period and phase shift of y 3 cos 2x and graph one complete period

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