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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.

Trig graphs model repeating change. A complete graph description separates amplitude, period, phase shift, midline, and tangent asymptotes so students can interpret and write equations from features.

What you'll learn

Why it matters: Tides, temperature, sound, daylight, wheels, and alternating current are modeled with trig graphs.

Worked example

Problem. Graphing Sine and Cosine: Find the phase shift of y = sin(x - pi/2).

  1. The form x - h shifts right by h.
  2. Here h = pi/2.
  3. State the direction and amount.

Answer: right pi/2

Practice problems

1. Graphing Sine and Cosine: Find the amplitude of y = 3sin(x) + -2.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Graphing Sine and Cosine: Find the amplitude of y = 3sin(x) + -2.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Amplitude is the absolute value of the outside coefficient.
  4. The coefficient is 3.
  5. The amplitude is 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. Graphing Sine and Cosine: Find the midline of y = 4cos(x) + -1.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Graphing Sine and Cosine: Find the midline of y = 4cos(x) + -1.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. The vertical shift sets the midline.
  4. The shift is -1.
  5. The midline is y = -1.
  6. Check the result by substituting or estimating: the response should match y = -1 and make sense in the original problem.

Answer: y = -1

3. Graphing Sine and Cosine: Find the period of y = sin(4x).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Graphing Sine and Cosine: Find the period of y = sin(4x).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. For sine, period = 2pi/|b|.
  4. Here b = 4.
  5. The period is 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. Graphing Sine and Cosine: Find the phase shift of y = sin(x - pi/2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Graphing Sine and Cosine: Find the phase shift of y = sin(x - pi/2).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The form x - h shifts right by h.
  4. Here h = pi/2.
  5. State the direction and amount.
  6. Check the result by substituting or estimating: the response should match right pi/2 and make sense in the original problem.

Answer: right pi/2

5. Graphing Sine and Cosine: Name one vertical asymptote of y = tan(x).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Graphing Sine and Cosine: Name one vertical asymptote of y = tan(x).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Tangent is undefined when cosine is zero.
  4. cos(pi/2) = 0.
  5. So x = pi/2 is an asymptote.
  6. Check the result by substituting or estimating: the response should match x = pi/2 and make sense in the original problem.

Answer: x = pi/2

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