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Step 14 · Trigonometry and precalculus topics

Graph sine and cosine as waves, read and set their amplitude and period, and shift them up, down and sideways to model things that repeat.

Sine and cosine waves

As the angle goes round the unit circle again and again, sin x rises and falls between −1 and 1. Its graph is a wave that repeats every 2π. Cosine is the same wave shifted π/2 to the left.

y = sin x (solid) and y = cos x (dashed) from 0 to 2π, about 6.28: the same wave, shifted.
x0π2\frac{\pi}{2}π3π2\frac{3\pi}{2}2π
sin x010−10
cos x10−101

Amplitude and period

y = A sin(Bx) stretches the wave. The amplitude |A| is the height above the middle line. The period, the length of one full wave, is 2π ÷ B.

Example: Amplitude and period

Find the amplitude and period of y = 3 sin(2x).

  1. The amplitude is 3.
  2. The period is 2π ÷ 2 = π.

AnswerAmplitude 3, period π

y = 3 sin(2x) reaches 3 and −3 and repeats every π, about 3.14.

Example: Writing an equation

A wave has amplitude 5 and period 4π. Write it as a sine function.

  1. A = 5.
  2. 2π ÷ B = 4π, so B = 12\frac{1}{2}.

Answery=5sin⁡(12x)y = 5\sin\left(\tfrac{1}{2}x\right)

Shifts

y = A sin(B(x − h)) + k moves the wave h to the right and k up. Its middle line moves to y = k.

Example: Reading the shifts

Describe y=2sin⁡(x−π4)+1y = 2\sin\left(x - \frac{\pi}{4}\right) + 1.

  1. The amplitude is 2 and the period is 2π.
  2. It is shifted π4\frac{\pi}{4} to the right and 1 up, so it runs between −1 and 3.

AnswerMidline y = 1, highest 3, lowest −1

Example: Tides

The water depth in a harbor is h=1.5cos⁡(πt6)+4h = 1.5\cos\left(\frac{\pi t}{6}\right) + 4 meters, t hours after midnight. Find high tide, low tide and the period.

  1. The midline is 4 and the amplitude 1.5: high tide 5.5 m, low tide 2.5 m.
  2. The period is 2π÷π6=122\pi \div \frac{\pi}{6} = 12.

AnswerHigh 5.5 m, low 2.5 m, every 12 hours

Try it yourself

Work each one on paper first, then open the answer to check.

  1. What is the period of y = sin(4x)?

    Show answer

    π2\frac{\pi}{2}

  2. What is the amplitude of y = −6 cos x?

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    6

  3. What are the largest and smallest values of y = 3 sin x + 2?

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    5 and −1

  4. A sine wave y = sin(Bx) has period 10. What is B?

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    π5\frac{\pi}{5}

  5. How is y = sin(x + π) shifted from y = sin x?

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    π to the left

  6. A wave goes from a high of 9 to a low of 1. What are its amplitude and midline?

    Show answer

    Amplitude 4, midline y = 5

What next?