|sin(α)||opposite / hypotenuse|
|cos(α)||adjacent / hypotenuse|
|tan(α)||opposite / adjacent|
|α, adjacent||opposite||= adjacent * tan(α)|
|hypotenuse||= adjacent / cos(α)|
|α, opposite||adjacent||= opposite / tan(α)|
|hypotenuse||= opposite / sin(α)|
|α, hypotenuse||adjacent||= hypotenuse * cos(α)|
|opposite||= hypotenuse * sin(α)|
For a concrete example, let's say I'm computing the dimensions of a taper for a machine part. Imagine that the base form is a cylinder, and from this a taper will rise. If the included angle of the taper is 3 degrees (which would give a non-releasing taper), and if I draw a sectional view of the taper, the acute angle for each right triangle defining the taper outside of the cylinder is 1.5 degree. Let's say also that I know how far along this cylinder the taper should extend; say 15mm
First (and typically most usefully) I want to know the length of the side opposite the angle. For the taper example, this would give the amount that the taper rises above the cylinder; the difference between the radius of the the taper at its large end and the radius of the cylinder there.
Intuitively, this checks out: A slight taper such as this won't rise much, and the hypotenuse of a very skinny triangle such as this isn't going to be much longer than the leg adjacent to its acute angle.
What if I don't know the adjacent leg, but do know the opposite? (In terms of the taper example, this would imply that I know how high/thick the taper should be, and I want to find out how long a taper I need to cut to achieve this.)
What about the hypotenuse? Just as, in the previous section, the cosine function related the side I knew (then, the adjacent side) and the hypotenuse, so here the sine function relates the side I know (now, the opposite) and the hypotenuse. So:
All portions of this document not noted otherwise are Copyright © 2008 by David M. MacMillan and Rollande Krandall.
Circuitous Root is a Registered Trademark of David M. MacMillan and Rollande Krandall.
This work is licensed under the Creative Commons "Attribution - ShareAlike" license. See http://creativecommons.org/licenses/by-sa/3.0/ for its terms.
Presented originally by Circuitous Root®
Select Resolution: 0 [other resolutions temporarily disabled due to lack of disk space]