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The Sine Function

Section 13.4. The Sine Function. The sine function , y = sin θ , matches the measure θ of an angle in standard position with the y -coordinate of a point on the unit circle. This point is where the terminal side of the angle intersects the unit circle. The Sine Curve Graph. maximum.

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The Sine Function

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  1. Section 13.4 The Sine Function

  2. The sine function, y = sin θ, matches the measure θ of an angle in standard position with the y-coordinate of a point on the unit circle. • This point is where the terminal side of the angle intersects the unit circle.

  3. The Sine Curve Graph maximum zero zero zero minimum

  4. The graph on the last slide is called the parent function of the sine function. • This means that: • 1. it has not been stretched or compressed, • 2. it has not been reflected over an axis, • 3. it has not been moved up/down or right/left. • Today we will look at the first two transformations.

  5. The sine curve is a periodic function. • A periodic function repeats a pattern of y-values at regular intervals. • One complete pattern is a cycle. • The period of a function is the horizontal length of one cycle. • The period of the parent function sine curve is 2π.

  6. Finding the Period of y = sin bθ • The period of a sine curve in the form y = sin bθ is • b is the number of cycles that can be drawn from 0π and 2π.

  7. Example • Find the period of the following sine curves and tell how many cycles will be drawn from 0 to 2π: • a. y = sin 4θ

  8. b. y = sin 8θ

  9. Here is the graph for each of the problems from the example. • a. y = sin 4θ

  10. b. y = sin 8θ

  11. Amplitude of a Sine Curve • The amplitude of a periodic function is half the difference between the maximum and minimum values of the function. • The amplitude of the parent function sine curve is 1.

  12. Finding the Amplitude of y = a sin bθ • The amplitude of y = a sin bθ is a in the equation.

  13. Example • Find the amplitude and period for each sine. Then write an equation for each curve

  14. a.

  15. To find the equation for the sine curve, you must find b. Remember b is the number of cycles from 0π to 2π. • The equation is y = a sin bθ.

  16. b.

  17. c.

  18. Reflection Across the x-axis • If a in the equation y = a sin bθ is negative, then the sine curve is reflected across the x-axis.

  19. Example • Sketch one cycle of the graph of the sine curve. • a. y = 3 sin 2θ • 1. The amplitude of the curve is 3. Mark this on the y-axis. • 2. The period is

  20. 3. Divide the period into fourths to find the hash marks for the graph. • 4. The pattern for a sine curve is • zero-maximum-zero-minimum-zero

  21. b. • 1. What is the amplitude of the sine curve? • amplitude = 2 • 2. What is the period? • period = 6π

  22. 3. What would you divide x-axis into? • 4. What is the pattern for a sine curve? • zero-maximum-zero-minimum-zero

  23. c.y = -4 sin 3θ • 1. What is the amplitude of the sine curve? • amplitude = 4 • 2. What is the period?

  24. 3. What would you divide x-axis into? • 4. What is the pattern for a sine curve? • Since there is a negative sign at the beginning of the equation, we reflect the graph over the x-axis. • zero-minimum-zero-maximum-zero

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