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ss_5_6 - Section 5.6 Sinusoidal Graphs This section...

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Section 5.6 Sinusoidal Graphs This section contains analysis of sine and cosine as periodic functions. Definition. If f ( t ) = A sin ωt and g ( t ) = A cos ωt , then the number | A | is called the amplitude of f and g , and the number T = 2 π ω , ω > 0, is called the period of f and g . • | A | Amplitude of A sin ωt , A cos ωt T = 2 π ω ( ω > 0) Period of A sin ωt , A cos ωt Note. The above formula for the period can be obtained as follows. The functions f ( t ) = sin ωt and g ( t ) = cos ωt complete one full cycle when ωt changes from ωt = 0 to ωt = 2 π . This change in ωt occurs when t changes from t = 0 to t = 2 π ω . Example. Find the amplitude and period of y = - 2 sin 3 t . Solution. y = - 2 sin 3 t is of form y = A sin ωt with A = - 2 and ω = 3. Hence the amplitude is | A | = 2 and the period is T = 2 π 3 . The graph of this function is shown below. –2 –1 0 1 2 y –3 –2 –1 1 2 3 t y = - 2 sin3 t Example. Find the amplitude and period of y = 3 cos( - 2 t ). Solution. Write y = 3 cos( - 2 t ) in the form y = 3 cos(2 t ). (The last step is permissible because the cosine is an even function.) The amplitude is | A | = 3 and the period T = 2 π 2 = π . The graph of this
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