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Period and Frequency

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

For a function y=sin(bx), the coefficient b horizontally stretches or compresses one full cycle, giving a period of T=2pi/|b| radians. Larger |b| means a shorter period (more cycles), and you can reverse the formula as b=2pi/T to recover b from a known period. Frequency is the reciprocal of period, f=1/T, which also equals |b|/(2pi) — it counts how many cycles fit in an interval of length 1.

What you'll learn

Why it matters: Sound pitch, AC electrical signals, and ocean waves are all described by frequency and period: a higher frequency (larger b) means more cycles packed into each second, which is why a 440 Hz tuning-A vibrates far faster than a deep bass note.

Worked example

Problem. Find the period of y=sin(4x).

  1. Identify b=4, so |b|=4
  2. Apply the period formula T=2pi/|b|
  3. T=2pi/4=pi/2

Answer: pi/2

Practice problems

1. Find the period of y=sin(2x).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the period of y=sin(2x).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Here b=2, so |b|=2
  4. Apply T=2pi/|b|
  5. T=2pi/2=pi
  6. Check the result by substituting or estimating: the response should match pi and make sense in the original problem.

Answer: pi

2. Find the period of y=sin(4x).

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

3. In the formula for the period of y=sin(bx), what does b control?

Choices: How fast the graph cycles (horizontal stretch/compression) · The vertical height of the wave · The up-or-down shift of the wave · The left-or-right shift of the wave

Show solution
  1. The period is T=2pi/|b|
  2. A larger |b| shrinks the period, packing in more cycles
  3. So b governs horizontal compression and how fast it cycles

Answer: How fast the graph cycles (horizontal stretch/compression)

4. Find the period of y=sin(3x). Give the exact answer.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the period of y=sin(3x). Give the exact answer.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Here b=3
  4. T=2pi/|b|=2pi/3
  5. T=2pi/3
  6. Check the result by substituting or estimating: the response should match 2pi/3 and make sense in the original problem.

Answer: 2pi/3

5. Find the period of y=sin((1/2)x).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the period of y=sin((1/2)x).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Here b=1/2
  4. T=2pi/(1/2)=2pi*2
  5. T=4pi
  6. Check the result by substituting or estimating: the response should match 4pi and make sense in the original problem.

Answer: 4pi

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