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Graphs of Sine and Cosine

A free Algebra II lesson from the “Trigonometry and Modeling” unit, with a worked example and practice problems including step-by-step solutions.

Sine and cosine graphs repeat in waves. Amplitude measures distance from the midline to a peak, and period is the length of one complete cycle.

What you'll learn

Why it matters: Tides, sound waves, heartbeats, and AC voltage all graph as sine and cosine waves. Amplitude controls the height, period controls how long one full wave is, and the phase shift slides the wave horizontally.

Worked example

Problem. For y = 3sin(x), what is the amplitude?

  1. The amplitude is the absolute value of the coefficient on sine.
  2. |3| = 3.
  3. The graph rises 3 units above and below the midline.

Answer: 3

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. For y = 5sin(x), what is the amplitude?

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  1. Amplitude is |5|.

Answer: 5

2. For y = -2cos(x), what is the amplitude?

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  1. Amplitude is distance, so use absolute value.

Answer: 2

3. What is the period of y = sin(x)?

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  1. Basic sine has period 2pi.

Answer: 2pi

4. For y = sin(x) + 4, the midline is...

Choices: y = 4 · y = 1 · x = 4 · y = -4

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  1. The +4 shifts the graph up.

Answer: y = 4

5. What is sin(0 degrees)?

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  1. At 0 degrees, the unit-circle point is (1, 0).
  2. Sine is the y-coordinate.
  3. The y-coordinate is 0.

Answer: 0

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