From wsanford@wsanford.com Sat Jul 30 08:37:04 2005 Date: Sat, 30 Jul 2005 08:33:46 -0400 (EDT) From: Walter Sanford To: Walter Sanford Subject: Re: tangent (fwd) Original msg. from PSW. ================================================================= 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/ ================================================================= ---------- Forwarded message ---------- Date: Sat, 23 Jul 2005 22:07:42 -0400 From: Phil Wherry To: Walter Sanford Subject: Re: tangent You're closer to the answer than you think. Since we're dealing with a unit (i.e. radius=1) circle here, the tangent of an angle is the length relationship between the radius line of the circle (which is always 1; the only thing that varies is the angle with respect to the X axis) and the tangent line between the crossing point and the X axis. The tangent of 45 degrees is 1 because the tangent line and the radius line are the same length. When you talk about something in radians, you're telling someone that you're talking about an angle: specifically, the angle that would be formed by taking a unit circle (radius=1) and stretching a string of a particular length from the point where the circle crosses the X axis (1,0) along the curve of the circle to the line that forms the other half of the angle. Let's consider an example: 1.57 radians. Since we know that a unit circle has a circumference of roughly 6.28 (2 * pi * r, and r=1 because it's a unit circle), we can tell that 1.57 radians is one quarter of a circle, or 90 degrees. In other words, take a string of 1.57 units in length and start wrapping it around a circle with radius 1. When you run out of string, you'll be at the one-quarter (or 90 degree) mark. Phil == Walter Sanford wrote: >Phil, > >FWIW, I realized how dumb my second question is immediately after I posted >it! Even if the answer is true in the case of a 45deg angle, it is >certainly not true for all angles. For a minute, I thought I had a great >insight into the relationship between radians and the TAN function -- >NAWWW! > >Walter > >-- >On Sat, 23 Jul 2005, Walter Sanford wrote: > > > >>On Thu, 21 Jul 2005, Phil Wherry wrote: >> >>I made some drawings & visited the Wikipedia website -- it's beginning to >>make sense! So, TAN45 is 1 radian, right? >> >> >> >>>To come back to an earlier topic: you'll find when you do the drawings >>>that radians will probably make more sense as an angular measure, too. >>>If you were to wrap a string around that same unit circle starting from >>>the X axis until you reach the intersection point between your line and >>>the circle, the length of the string would be the angular measure in >>>radians. Note, too, that reduces everything in trig to distances; no >>>longer do you have to explain angular measures as these arbitrary units: >>>everything is just a length, whether straight-line or curved! >>>Phil >>> >>> >>In the preceding example, would the length of the arc traced between the >>tangent point and the x-axis be same as the length of the TAN side of the >>triangle? >> >> >> >>>Walter Sanford wrote: >>> >>> >>> >>>>Phil, >>>> >>>>Thanks for the quick (& detailed!) reply! I've said it before, you would >>>>be a *great* math teacher. I need to make some drawings, but your >>>>explanation makes sense after a quick once-over. >>>> >>>>================================================================= >>>>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/ >>>>================================================================= >>>> >>>> >>>> >>>> >>================================================================= >>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/ >>================================================================= >> >> >> >> > >================================================================= >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/ >================================================================= > >