Trigonometry

Let's explore the sine a little more and see what the graph looks like. To help us get there, let's place the right triangle in a unit circle (a circle with a radius of 1).

This means the hypotenuse will always be 1. If the point on the circle the hypotenuse touches is $(x,y),$ what is the sine of the marked angle?

Hint: Which two sides of the triangle do we use to define the sine of the angle?

Graphing the Sine

Based on the previous question, we can think of taking the sine as the equivalent of keeping track of the $y$-coordinate of a point that rotates around a unit circle.

What does the graph of $f(x)=\sin(x)$ look like?

Graphing the Sine

Since the sine graph is made by rotating around a unit circle, it cycles every time the full circle is complete.

Which of the answer choices makes a graph identical to that of $f(x)=\sin(x) ?$

Hint: The transformation $f(x - h)$ shifts the graph right by $h .$

Graphing the Sine

By turning the graph of the sine into a wave, we have a representation that matches how radio waves, sound waves, light waves, and many more types of waves act.

Pitch is a way to position notes in music that is "lower" or "higher"; the graph above shows a sound wave that starts low in pitch and gradually increases in pitch.

The graph below shows two sound waves. What is true?

(Note: You can hear these waves in the answer!)

Graphing the Sine

Waves can be combined to form increasingly complex sounds; even the most complex, multi-instrument music amounts to just a single sound wave that comes from adding multiple waves. In other words, no matter how complex, a sum of sine waves will be able to sound like literally anything.

Later in this course, you’ll learn a lot more about sine (and cosine) waves and about how to work with them algebraically as well as intuitively. If you like, you can play with a pair of sine waves below and see what they look like when combined (the meaning of the different variables will be explored during the course).

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