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13.4 – The Sine Function

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**I. Interpreting Sine Functions**

The sine function, y = sin θ, matches the measure θ of an angle in standard position with the y – coordinate on the unit circle. The periodic sine function has one cycle every 360° or 2π Hence the sine of any degree greater than 2π, then the cycle is repeated.

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**The x-axis or domain will be thought of in terms of theta, θ**

When graphing you have to pay attention to the domain of theta. Trig Cycles are repeated once every 360 degrees or 2π

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**The General Sine Curve:**

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**Example 1: Graph, using a table the values of the sine curve in the domain**

0 ≤ θ ≤ 2π. Use only the coordinates on the θ and y axes.

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You can vary the period and amplitude of the sine curve to get different curves, either “extended” or more “frequent” Done my multiplying the function by a constant, and theta by a constant

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Y = 3 sin θ

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Y = sin 2θ

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**II. Properties of the Sine Function**

y = a sin bθ ІaІ = the amplitude (the highest and lowest points on the curves.) b = the number of cycles the curve makes from 0 to 2π 2π / b = the period of the cycle

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**Steps to Graphing y = a sin bθ**

Step 1: draw your reference curve Step 2: identify the domain and set up your graph. Step 3: identify the amplitude and document Step 4: identify the number of cycles in the domain that you have to graph Step 4: identify the period you need to create one cycle Step 5: Alter graph in needed in increments of 4 units on the theta axis to meet the criteria of the total cycles.

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Example 2: how many cycles does the following have in the domain from 0 to 2π? Give the amplitude and period as well. A) y = -5 sin 2θ B) f(x) = .5 sin .5θ

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**Example 3: Graph the following. (check on the calculator)**

A) y = -2 sin 2θ, for 0 < θ < 2π B) y = 5 sin θ, for 0 < θ < 2π C) y = ½ sin .5θ, for 0 < θ < 2π

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