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Graphs of Sine Curves Graph Transformations of the Sine Function Graph Transformations of the Cosine Function Determine the Amplitude and Period of Sinusoidal Functions Graph Sinusoidal Functions: y= A sin Bx Find the Equation for a Sinusoidal Graph

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Periodic Functions If we add or subtract integral multiples of 2π to θ, the trigonometric values remain unchanged. This is for all θ A function f is called periodic if there is a positive number p such that, whenever θ is in the domain of f so is θ+p and f(θ + p) = f(θ) If there is a smallest such number p, is the smallest value is called the fundamental period of f

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Periodic Properties sin ( θ + 2π) = sin θ cos (θ +2π) = cos θ

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The graph of y = sin x Since the sine function has a period of 2π we only need to graph on the interval [0, 2π] The graph is one period or one cycle of the graph of y= sin x

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Properties of the sine function 1.The domain is the set of all real numbers 2.The range consists of all real numbers from -1 to 1 3.The sine function is an odd function, as the symmetry of the graph with respect to the origin indicates 4.The sine function is periodic with a period of 2π 5.The x intercepts are ……… -2π, -π, 0, π, 2π, 3π, …….. 6.The maximum value is 1 and occurs at x =.. and the maximum of -1 occurs at x=

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y = A sin Bx The number |A| is called the amplitude Amplitude = Period =

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The graph of y = cos x The cosine function has a period of 2π

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Properties of the cosine function 1.The domain is the set of all real numbers 2.The range consists of all real numbers from -1 to 1 3.The cosine function is an even function, as the symmetry of the graph with respect to the y axis indicates 4.The cosine function is periodic with a period of 2π 5.The x intercepts are 6. The maximum value is 1 and occurs at x =.-2π, 0, 2π,4π and the maximum of -1 occurs at x=.. –π,π,3π, 5π

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Sinusoidal Graphs Shift the graph of y = cos x to the right to obtain the graph of y= cos (x - ) because of the similarity of sine and cosine function we refer to them as sinusoidal graphs

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Graphs of Other Trigonometric Functions 4.6

Graphs of Other Trigonometric Functions 4.6

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