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Precalculus Warm-up:

Unit Circle Quiz Tuesday (ALL Quadrants) No Calculator. Precalculus Warm-up:. Find the exact values for each. Answers. Main events:. Objective: Graph sine and cosine functions. 4.5 Digital Lesson. Graphs of Trigonometric Functions. Sine and Cosine.

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Precalculus Warm-up:

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  1. Unit Circle Quiz Tuesday (ALL Quadrants) No Calculator PrecalculusWarm-up: Find the exact values for each.

  2. Answers

  3. Main events: • Objective: Graph sine and cosine functions.

  4. 4.5 Digital Lesson Graphs of Trigonometric Functions Sine and Cosine

  5. 2. The range is the set of y values such that . 5. Each function cycles through all the values of the range over an x-interval of . Properties of Sine and Cosine Functions Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 6. The cycle repeats itself indefinitely in both directions of thex-axis.

  6. x 0 sin x 0 1 0 -1 0 y = sin x y x Graph of the Sine Function Sine Function To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts. Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

  7. x 0 cos x 1 0 -1 0 1 y = cos x y x Graph of the Cosine Function Cosine Function To sketch the graph of y = cos x first locate the key points.These are the maximum points, the minimum points, and the intercepts. Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

  8. x 0  2 3 0 -3 0 3 y = 3 cos x max x-int min x-int max y (0, 3) ( , 3) x ( , 0) ( , 0) ( , –3) Example: Sketch the graph of y = 3 cos x on the interval [–, 4]. Example: y = 3 cos x Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.

  9. y y = 2 sin x x y = sin x y = sin x y = –4 sin x reflection ofy = 4 sin x y = 4sin x The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. Amplitude amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| <1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis.

  10. For b 0, the period of y = a sin bx is . For b 0, the period of y = a cos bx is also . period: period: 2 y x y period: 2 x period: 4 The period of a function is the x interval needed for the function to complete one cycle. Period of a Function If b > 1, the graph of the function is shrunk horizontally. If 0 < b < 1, the graph of the function is stretched horizontally.

  11. NOTES 4.5 & 4.6 Translations of Sineand Cosine

  12. Start graph on axis Start graph at amplitude.

  13. Example 1 graph a) y = sin x

  14. Graph 2 full periods (cycles)

  15. x y Graph 2 full periods (cycles)

  16. Example 1 (b)

  17. Graph 2 full periods (cycles)

  18. Ex 1 c)

  19. Graph 2 full periods (cycles)

  20. Example 2 Graph: a) y = cos x

  21. Graph 2 full periods (cycles)

  22. EX 2 b) y = 3cos 4x-π

  23. Graph 2 full periods (cycles)

  24. Classwork p. 326 # 5-10, 19-37 odd, 39, 43, 57, 59 Homework p. 326 [ Evens] #12-18, # 32-36, 50-54

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