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
- Identify amplitude and period
- Compare sine and cosine graphs
- Use basic sinusoidal models
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?
- The amplitude is the absolute value of the coefficient on sine.
- |3| = 3.
- The graph rises 3 units above and below the midline.
Answer: 3
Practice problems
1. For y = 5sin(x), what is the amplitude?
Show solution
- Warm-up: First identify exactly what the question is asking: For y = 5sin(x), what is the amplitude?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Amplitude is |5|.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
2. For y = -2cos(x), what is the amplitude?
Show solution
- Warm-up: First identify exactly what the question is asking: For y = -2cos(x), what is the amplitude?
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- Amplitude is distance, so use absolute value.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
3. What is the period of y = sin(x)?
Show solution
- Core Practice: First identify exactly what the question is asking: What is the period of y = sin(x)?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Basic sine has period 2pi.
- Check the result by substituting or estimating: the response should match 2pi and make sense in the original problem.
Answer: 2pi
4. For y = sin(x) + 4, the midline is...
Choices: y = 4 · y = 1 · x = 4 · y = -4
Show solution
- Challenge: First identify exactly what the question is asking: For y = sin(x) + 4, the midline is...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The +4 shifts the graph up.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: y = 4
5. What is sin(0 degrees)?
Show solution
- Review: First identify exactly what the question is asking: What is sin(0 degrees)?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- At 0 degrees, the unit-circle point is (1, 0).
- Sine is the y-coordinate.
- The y-coordinate is 0.
- Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.
Answer: 0
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