How to Find the Amplitude of a Sine Wave

Step-by-step solution

1. Match the function to the form y = A sin(bx).

   $$A = 5,\quad b = 2$$

   Why: The number in front of sine is A, and the number multiplying x is b.

2. Find the amplitude.

   $$|A| = |5| = 5$$

   Why: The amplitude is the height of the wave from its middle to its peak, so it is always positive.

3. Find the period with T = 2 pi over b.

   $$T = \frac{2\pi}{2} = \pi$$

   Why: A larger b makes the wave repeat faster and shortens the period.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $y = 3\cos(x)$

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   $\text{amplitude } 3,\quad \text{period } 2\pi$

2. 2

   $y = 5\sin(4x)$

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   $\text{amplitude } 5,\quad \text{period } \frac{\pi}{2}$

3. 3

   $y = 2\cos(\pi x)$

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   $\text{amplitude } 2,\quad \text{period } 2$

4. 4

   $y = 7\sin\left(\frac{x}{2}\right)$

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   $\text{amplitude } 7,\quad \text{period } 4\pi$

5. 5

   $y = 6\cos(3x)$

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   $\text{amplitude } 6,\quad \text{period } \frac{2\pi}{3}$