Exploring Amplitude in Sine and Cosine Waves

When we look at trigonometric functions, one exciting thing to understand is how the amplitude, or height, changes how sine and cosine waves look. Let’s make this easier to grasp!

What is Amplitude?

Amplitude is simply the height of the wave. It shows how far the wave goes above and below its center line. For the basic sine and cosine functions:

• The amplitude is usually 1.
• This means the waves go up to 1 and down to -1.

Changing the Amplitude

When we change the amplitude to a new value, let’s call it A, we update our sine and cosine functions like this:

• Sine: ( y = A \sin(x) )
• Cosine: ( y = A \cos(x) )

How Amplitude Changes Look

1. Bigger Amplitude:

   • If we double the amplitude to 2, the new functions would be:
     • ( y = 2 \sin(x) )
     • ( y = 2 \cos(x) )
   • Graph Changes: Now the highest point of the wave is 2 and the lowest point is -2. The wave looks taller and stretched out when we graph it.

2. Smaller Amplitude:

   • If we reduce the amplitude to 0.5, the functions become:
     • ( y = 0.5 \sin(x) )
     • ( y = 0.5 \cos(x) )
   • Graph Changes: The peaks and valleys of the waves are now at 0.5 and -0.5. This makes the waves shorter and softer compared to the original sine and cosine waves.

In Summary

Playing around with the amplitude of sine and cosine waves changes how they look a lot! Remember:

• A bigger amplitude means taller waves.
• A smaller amplitude means shorter waves.

This shows us not just how beautiful trigonometric functions can be, but also how flexible they are. It’s fun to draw different versions of these waves and see how they change when we tweak the amplitude!