From psw@wherry.com Thu Jul 21 12:23:07 2005 Date: Thu, 21 Jul 2005 09:21:29 -0400 From: Phil Wherry To: Walter Sanford Subject: Re: tangent [ The following text is in the "ISO-8859-1" character set. ] [ Your display is set for the "US-ASCII" character set. ] [ Some characters may be displayed incorrectly. ] It's because basically all of the trig functions are defined in terms of a unit (radius=1) circle. To visualize this: * Think of a circle with radius 1, centered on the origin. * Draw a line from the center of the circle out to some point on the circle. Measure the angle between that line and the X axis (counterclockwise from the axis). * Now draw a tangent line to the circle at the point where the line you drew first intersects the circle. Note that be definition this is going to be perpendicular to your original line. * Measure the length of the tangent line between the point where it touches the circle and the X axis. This is the trigonometric tangent. Here are some examples: * First, let's try a 45-degree angle. Draw your circle, then the 45-degree angle (measure this counterclockwise from the X axis; this is how all angles are measured mathematically). Now draw your tangent line. Since it's at a right angle to the line you drew, it too will be at a 45-degree angle with respect to the X-axis. It's fairly obvious when you draw this that the segment between the intersection point and the X axis is of equal length to the line you originally drew (which we know to be of length 1 since we started with a unit circle). Therefore, tangent(45 degrees) equals 1. * Now let's try a 0-degree angle. Draw the circle. Your line in this case is the X axis between (0,0) and (1,0). Draw the tangent line; in this case it will be a perfectly vertical line where X=1. When you try to measure the distance between the point (1,0) and the point where the tangent line intersects the X axis (1,0), you'll discover there's no distance at all--it's just a point. So tangent(0 degrees) equals 0. * Let's look at the 90-degree angle. Once again, draw your circle. Your line in this case extends up from (0,0) to (1,0). Then draw your tangent line at right angles to it. You'll run into trouble when you try to measure the distance between the intersection point and the X axis, because the tangent line is parallel to the X axis and therefore never crosses. That's why tangent(90 degrees) is undefined -- you'll get an error if you try to compute it. * So what about something close to 90 degrees? Let's try 89 degrees. The line you draw first from the origin is almost but not quite vertical. Draw the tangent line. Because it's close to but not quite parallel to the axis, the distance between the intersection point and the X axis should be fairly big. And it is: tangent(89 degrees) equals roughly 57.3. Related asides: the line you draw intersects the circle at some point (x,y). The value of y is the sine of the angle, and the value of x is the cosine. Once you've drawn the examples above (do these first, seriously), go have a look at: http://en.wikipedia.org/wiki/Unit_circle Phil Walter Sanford wrote: Phil, If the tangent of circle is a line perpendicular to the radius that touches ONLY one point on the circle, then why is the same word used for angles (TAN)? I searched the 'net but did not find an answer. ================================================================= Walter Sanford, Director Carl Sandburg Middle School Center for Sky Awareness 8428 Fort Hunt Road Sandburg Planetarium Alexandria, VA 22308 Fairfax County Public Schools Work: 703-799-6169 -6197 (fax) E-mail: wsanford@wsanford.com Home: 703-765-9392 AMS Project ATMOSPHERE Atmospheric Education Resource Agent & Water in the Earth System (WES) Resource Teacher SCSA, Geosystems, & Camp T-Equity - URL: http://www.wsanford.com/ =================================================================