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The Definitions of Sine and Cosine - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The right triangle definitions of sine and cosine only apply to acute angles, so a more complete definition is needed. The point where the terminal side intersects the unit circle (x, y) is the basis for this definition. Since the radius (and therefore hypotenuse of the right triangle) is 1, the denominators cosine=adjacent/hypotenuse and sine=opposite/hypotenuse are also 1. Thus, the sine definition is y=sine and x=cosine.

I want to talk about something really important the definition of sine and cosine. Now you might remember from Geometry the right triangle definitions of sine and cosine it starts with a right triangle and we'll label the 3 sides x, y and z the acute angle here is theta and this is a right angle. We defined cosine of theta to be the side adjacent to theta divided by the hypotenuse. And by adjacent we mean the side that's next to theta, this is the hypotenuse the long side of the right triangle and so that means x over z. The sine is defined as the side opposite theta y over the hypotenuse, so y over z.
The problem with this definition is that it only works for acute angles. So that means that theta has to be between 0 and 90 degrees right or else this triangle won't make sense, so one of the things we do in pre-calculus is extend this definition so that it includes all angles. Okay this is what an angle looks like in standard position in standard position you draw the angle so that its vertex is on the origin in a coordinate plane. This is the initial side, this is the terminal side and you can think of an angle as a rotation as if the terminal side was starting here and rotating through an angle of theta ending here.
Now add to that angle in standard position, the unit circle, the circle with radius 1 x squared plus y squared equals 1, is this is circle here. Now just to get your bearings, when you have a circle radius 1 it's going to pass through the point 1, 0 it'll pass through the point 0, 1 negative 1, 0 and 0 negative 1. Same angle, we want to mark the point where the terminal side intersects the unit circle. That point will have coordinates x, y we define cosine to be the x value and sine to be the y value. This point of course will be unique it'll depend uniquely on the angle theta, so for different angle thetas you're going to get different sine and cosine values but this idea here will allow us to measure sine and cosine for any angle at all. It'll work for the acute angles when theta is in the first quadrant here.
It'll work for 0 degrees, 90 degrees and any other angle, so this is the power of the unit circle definitions is that they work for all angles we'll be using these for the rest of the trigonometry course.

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