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# Sine and cosine come from circles

Sine and cosine can be generated by projecting the tip of a vector onto the y-axis and x-axis as the vector rotates about the origin.  Created by Willy McAllister.

## Want to join the conversation?

• Will drawing a circle with a bigger radius allow us to find angles that have sine or cosine values that are greater than 1?
(1 vote)
• Clever idea, but no. Trig functions (sine, cosine, tangent) are defined as ratios. Sine is "opposite over hypotenuse" (the SOH of SOHCAHTOA). When we draw the triangle inside a unit circle the hypotenuse is automatically 1 at any angle. That means the sine of an angle is simply the length of the "opposite" leg of the triangle (opposite / 1). If you make the circle radius = 2 it makes both O and H twice as long, but the ratio stays the same. Nice try. I like the way you think.
• Hello, we are starting the curves of sine and cosine at school, is this what I need to know for an understanding of it, I mean is this the basics for starting the sine and cosine graphs/curves? Thank you
(1 vote)

When sine and cosine are first revealed in Trigonometry class they are taught as ratios of the the sides of right triangles. The memorization tool is "SOA CAH TOA".

If you fit the right triangle inside a circle, you can slide the tip of the triangle around the circle and think about sin and cos of a point somewhere along the circle. It all means the same thing, just a slightly different mental picture.

Using this circle perspective is really handy. In the next few videos we let the point on the circle move as time goes by, so you get this orbiting point. That turns out to be a super useful way to think about Signals. (sound, radio waves, light)

Meanwhile, in Trig class you will use the right triangle viewpoint to prove all sorts of useful properties and identities of trig functions.