Understanding the Unit Circle and Trigonometry Functions

[Peter G.]

I am learning about the unit circle and I am a bit confused.

So, I have my circle drawn with radius 1 and I sketched a right angled triangle inside it so that the hypothenuse has a length of 1.

I think what is making me confused is the meaning of sine, cosine and tangent.

They are functions of an angle x that enable me to find the length of a side of a triangle provided I have an angle and one length? In this case, with the hypothenuse being 1, it will be equal to the function of the angle? Is that right?

And, finally in the case of tangent for example, I am not finding the length am I?

Thanks,
Peter G.

 

[Mark44]

Where your triangle intersects the unit circle, the coordinate are (x, y). The x coordinate is the cosine of the angle; the y coordiante is the sine of the angle.

If you extend a vertical line up from the point (1, 0), and extend the hypotenuse of your triangle, they intersect at a point (1, y). The y coordinate is the tangent of your angle.

 

[HallsofIvy]

As Mark44 said, if you have your unit triangle with center at (0, 0) of a coordinate system and draw a line from (0, 0) to the circle at angle $\theta$ radians to the positive x-axis, then the coordinates of the point at which the line crosses the unit circle are $(cos(\theta), sin(\theta))$.

(radians because, strictly speaking, we are defining sine and cosine as functions of the arclength around the circumference of the unit circle, not as functions of the angle- but engineers like to always think of sine and cosine in terms of angles so we define radian measure to make the arclength the same as the angle.)

If you also draw the line from that point to the x-axis, you will have a right triangle with hypotenus of length 1, "opposite side" of length $sin(\theta)$ and "near side" of length $cos(\theta)$. Extending or shortening the lengths while maintaining the same angle gives "similar triangles" which always have the same ratios of side lengths so that, in general, $sin(\theta)$ is "opposite side divided by hypotenuse" and $cos(\theta)$ is "near side over hypotenuse".