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Graphing Sine and Cosine

A free Trigonometry lesson from the “Graphs of Trig Functions” unit, with a worked example and practice problems including step-by-step solutions.

The basic graph of y = sin(x) starts at the origin (0, 0), rises to a maximum of 1 at x = pi/2, returns to 0 at pi, drops to a minimum of -1 at 3pi/2, and returns to 0 at 2pi. The graph of y = cos(x) has the same wave shape but starts at its maximum (0, 1), so it leads sine by a quarter cycle. Multiplying by a (as in y = a*sin(x)) stretches the wave vertically so the max becomes a and the min becomes -a, while the zeros stay in the same x-locations.

What you'll learn

Why it matters: A swing or a child's playground swing traces a sine curve over time, while the up-and-down position of a clock's pendulum at its starting point follows a cosine curve, both showing the same smooth back-and-forth shape.

Worked example

Problem. For y = 4*sin(x) on 0 to 2pi, give the x-value where the maximum occurs and the maximum value there.

  1. The basic sine graph reaches its maximum at x = pi/2.
  2. Multiplying by 4 makes the maximum height equal to 4 instead of 1.
  3. So the maximum value 4 occurs at x = pi/2.

Answer: x = pi/2, maximum value 4

Practice problems

1. For the basic graph y = sin(x), what is the y-value at x = 0?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For the basic graph y = sin(x), what is the y-value at x = 0?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The sine graph passes through the origin.
  4. At x = 0, sin(0) = 0.
  5. So the y-value is 0.
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

2. For the basic graph y = cos(x), what is the y-value at x = 0?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For the basic graph y = cos(x), what is the y-value at x = 0?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The cosine graph starts at its maximum.
  4. At x = 0, cos(0) = 1.
  5. So the y-value is 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

3. On y = sin(x) over 0 to 2pi, at what x-value does the maximum value 1 occur?

Show solution
  1. Warm-up: First identify exactly what the question is asking: On y = sin(x) over 0 to 2pi, at what x-value does the maximum value 1 occur?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The sine wave rises after the origin.
  4. It peaks at x = pi/2.
  5. So the maximum 1 is at x = pi/2.
  6. Check the result by substituting or estimating: the response should match pi/2 and make sense in the original problem.

Answer: pi/2

4. On y = cos(x) over 0 to 2pi, at what x-value does the minimum value -1 occur?

Show solution
  1. Core Practice: First identify exactly what the question is asking: On y = cos(x) over 0 to 2pi, at what x-value does the minimum value -1 occur?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Cosine starts at 1 and decreases.
  4. It reaches its lowest point at x = pi.
  5. So the minimum -1 is at x = pi.
  6. Check the result by substituting or estimating: the response should match pi and make sense in the original problem.

Answer: pi

5. List the x-values where y = sin(x) equals 0 on the interval 0 to 2pi.

Show solution
  1. Sine crosses zero at the start, middle, and end of one cycle.
  2. Those crossings are at x = 0, pi, and 2pi.
  3. So the zeros are 0, pi, 2pi.

Answer: 0, pi, 2pi

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