File talk:Circle-trig6.svg

I'm not sure this is correct. I think the tangent line should be constructed from the initial arm to the secant line.

I don't see how using this construction you can arrive at the definition sec=1/cos or anything like that.

Furthermore, I haven't seen any sort of unified construction for all angles for the tangent, secant, cotangent, and co secant lines.

The image is correct. — Steven G. Johnson (talk) 23:23, 10 August 2012 (UTC)

How do you know it's correct? -TP

In particular, if this is correct, how would you arrive at the definitions tan=sin/cos or sec=1/cos?

I've been trying to work this out for several days and I haven't been able to figure it out. The best construction I can find is something like this http://mysite.du.edu/~jcalvert/math/trig.htm

I'll write out a detailed argument as to why I don't quite buy this construction over the next few days— Preceding unsigned comment added by TerryPhi86 (talk • contribs) 06:13, 11 August 2012 (UTC)

For example, consider the tan construction. Consider the right triangle ACE, whose hypotenuse AE we want to show is tan. The angle of corner E is 90°-θ (it is the complement of θ in triangle AOE). Since sine is opposite over hypotenuse, $\sin E=AC/AE$, and hence:

$\sin E=\sin(90^{\circ }-\theta )=\cos \theta ={\frac {AC}{AE}}={\frac {\sin \theta }{AE}},$

and it follows immediately that $AE=\sin \theta /\cos \theta =\tan \theta$. QED.

Given that AE=tan, the OE=sec result follows from the Pythagorean theorem applied to AOE: $(OE)^{2}=(OA)^{2}+(AE)^{2}=1+\tan ^{2}=\sec ^{2}$.

— Steven G. Johnson (talk) 22:06, 11 August 2012 (UTC)

Well, it took a little while, but I finally confirmed your construction. I did compass and ruler constructions for quadrants 1 and 3 and verified all the ratios met their definitions algebraically. This is an excellent construction, and I have much respect to you for creating it. What held me up for some time was not realizing that the tangent drawn at the terminal arm was at a right angle to the terminal arm. Thank you.

-TP — Preceding unsigned comment added by TerryPhi86 (talk • contribs) 11:37, 15 August 2012 (UTC)

An error exists, but not that one...

nitpicking...

The color of the cot line and its text are also inconsistent.

I have other minor improvements as well.

I moved the sec line moved to the top of the image, and the csc line to the right, completing a rectangle to allow cleaning up a bit, getting rid of the arrowheads and their associated dotted lines. I relabeled the points mnemonically because they were getting too numerous to be randomly assigned. O and A remain unchanged, but designated the points on the X and Y axis aligned vertically and horizontally with A as X and Y. I designated the 4 points intersecting where the unit circle intersects the axes as E,N,S and W for east, north, west, and south.

The point on the X axis that intersects the tangent line is designated as T, and the point on the Y axis that intersects the tangent line is designated as C after cotangent.

I also added the half versed functions within a unit square centered about the origin and designated the point on the half-unit circle inscribed inside of it that intersects the angle as H, and designated points U,D,L,R to those up, down, left, and right respectively on the unit square that are aligned horizontally or vertically with H.

I changed the colors, using double-lines of the same color, (with a thin yellow stripe down the middle) for co-functions, and used similar (adjacent) colors for hav-functions and unsaturated for ex-functions.

I also made use of CSS to clean up a lot of the code.

contribution

Here's my contribution.

   <svg viewbox="0 0 330 230" >
   <metadata>
       <title>Unit Circle</title>
       <subject>Trigonometry</subject>
       <description>A unit circle with the trigonometric functions labeled</description>
       <type>Image</type>
       <author>Steven G. Johnson, 德 Shaneyfelt, Limaner, Verdy p, Taktoa, Wikipedia</author>
       <format>image/svg</format>
   </metadata>
   
   <style>
   circle,rect,path,polyline {fill:none}
   .dash {stroke-dasharray:3,4;stroke:lightgrey;stroke-width:1.5;}
   text.co,g.co text{font-weight:bold;fill:none;stroke-width:.75;}
   line.co {stroke-width:3;stroke-opacity:.75;}
   line.co2 {stroke-width:1;stroke: yellow ;}
   text.co {fill: yellow }
   
   text.crd{fill:silver} 
   line.crd{stroke:silver}
   
   text.sin{fill:red} 
   line.sin,line.cos,text.cos{stroke:red}
   text.tan{fill:tan} 
   line.tan,line.cot,text.cot{stroke:tan}
   
   text.sec{fill:blue} 
   line.sec,line.csc,text.csc{stroke:blue}
   text.exs{fill: lightsteelblue} 
   line.exs,line.exc,text.exc{stroke: lightsteelblue}
   
   text.ver{fill:darkgreen} 
   line.ver,line.cvs,text.cvs{stroke:darkgreen}
   text.hav{fill: darkcyan}
   line.hav,line.hcv,text.hcv{stroke: darkcyan}
   	
   text.vcs{fill: darksalmon}
   line.vcs,line.cvc,text.cvc{stroke: darksalmon}
   text.hvc{fill:pink}
   line.hvc,line.hcc,text.hcc{stroke:pink}
   </style>
       
   <g style="fill:black">
       <text x="103" y="116" style="font-style:italic">θ</text>
       <text x="96" y="97">½</text>
       <text x="115" y="200">1</text>
        
       <text x="79" y="140">O</text>
       <text x="139" y="42">A</text>
       <text x="139" y="224">V</text>
       
       <text x="121" y="123">X</text>
       <text x="81" y="57">Y</text>
       
       <text x="190" y="140">E</text>
       <text x="82" y="33">N</text>
       <text x="8" y="140">W</text>
       <text x="82" y="212">S</text>
       
       <text x="83" y="20">C</text>
       <text x="312" y="140">T</text>
       <text x="312" y="20">F</text>  
       
       <text x="117" y="102">H</text>  
       <text x="145" y="87">R</text>  
       <text x="35" y="90">L</text>  
       <text x="104" y="185">D</text>  
       <text x="105" y="77">U</text>    
   </g>
   
   <line class="dash"  x1="134" y1="208" x2="134" y2="125"/>
   <g style="stroke:black">
       <circle cx="95" cy="125" r="91.7"/>
       <circle class="dash" cx="95" cy="125" r="45.85"/>
       <rect class="dash" x="49.15" y="79.15" width="91.7" height="91.7"/>
       <line id="OH" x1="95" y1="125" x2="114.5" y2="83.5"/>
       <line id="HA" class="dash"  x1="134" y1="42" x2="114.5" y2="83.5"/>
       <line id="OV" x1="95" y1="125" x2="134" y2="208"/>
       <path id="angle" d="M107,125C107,121 104,116 100,114"/>
       <polyline id="ortho" points="130,50 137,54 141,47"/> 
    </g>
    
    <line class="crd" x1="134" y1="42" x2="186.7" y2="125"/>
     
    <line class="sin" x1="134" y1="125" x2="134" y2="42"/>
    <line class="co cos" x1="95" y1="42" x2="134" y2="42"/>
    <line class="co2" x1="95" y1="42" x2="134" y2="42"/>
   
    <line class="tan" x1="134" y1="42" x2="313" y2="125"/>
    <line class="co cot" x1="95" y1="24" x2="134" y2="42"/>
    <line class="co2" x1="95" y1="24" x2="134" y2="42"/>
    
    <line class="sec" x1="95" y1="24.0" x2="312" y2="24.0"/>
    <line class="co csc" x1="312" y1="125" x2="312" y2="24"/>
    <line class="co2" x1="312" y1="125" x2="312" y2="24"/>
    
    <line class="exs" x1="186.7" y1="125" x2="313" y2="125"/>
    <line class="co exc" x1="95" y1="33.3" x2="95" y2="24"/>
    <line class="co2" x1="95" y1="33.3" x2="95" y2="24"/>
    
    <line class="ver" x1="134" y1="125" x2="186.7" y2="125"/>
    <line class="co cvs" x1="95" y1="42" x2="95" y2="33.3"/>
    <line class="co2" x1="95" y1="42" x2="95" y2="33.3"/>
    
    <line class="hav" x1="114.5" y1="83.5" x2="140.9" y2="83.5"/>
    <line class="co hcv" x1="114.5" y1="79.15" x2="114.5" y2="83.5"/>
    <line class="co2" x1="114.5" y1="79.15"  x2="114.5" y2="83.5"/>
    
    <line class="vcs" x1="3.3" y1="125" x2="134" y2="125"/>
    <line class="co cvc" x1="95" y1="216.7" x2="95" y2="42"/>
    <line class="co2" x1="95" y1="216.7" x2="95" y2="42"/>
    
    <line class="hvc" x1="49.15" y1="83.5" x2="114.5" y2="83.5"  />
    <line class="co hcc" x1="114.5" y1="83.5" x2="114.5" y2="170.85" />
    <line class="co2" x1="114.5" y1="83.5" x2="114.5" y2="170.85" />        
    
    <g style="font-size:15">
        <text class="crd" x="140" y="-100" transform="rotate(60)">crd</text>
       
        <text class="sin" x="63" y="-138" transform="rotate(90)">sin</text> 
        <text x="104" y="53" class="co cos">cos</text>
        
        <text class="tan" x="215" y="-22" transform="rotate(25)">tan</text>
        <text class="co cot" x="118" y="-22" transform="rotate(25)">cot</text>
        
        <text x="191" y="36" class="sec">sec</text>
        <text x="-85" y="305" class="co csc" transform="rotate(-90)">csc</text>
        
        <text x="225" y="122" class="exs">exs</text>
        <text x="59" y="32" class="co exc">exc</text>
        
        <text x="142" y="122" class="ver">ver</text>
        <text x="69" y="44" class="co cvs">cvs</text>
        
        <text x="142" y="96" class="hav">hav</text>
        <text x="100" y="65" class="co hcv">hcv</text>
        
        <text x="55" y="122" class="vcs">vcs</text>
        <text x="-109" y="90" class="co cvc" transform="rotate(-90)">cvc</text>
      
        <text x="70" y="75" class="hvc">hvc</text>
        <text class="co hcc" x="130" y="-119" transform="rotate(90)">hcc</text>
    </g>  
  </svg>

and a table to go with it:

A͡E

=

arc

AE

=

crd

=

chord

AX

=

sin

=

sine

AY

=

cos

=

cosine

AT

=

tan

=

tangent

AC

=

cot

=

cotangent

CF

=

sec

=

secant

TF

=

csc

=

cosecant

ET

=

exs

=

exsecant

NC

=

exc

=

excosecant

XE

=

ver

=

versine

YN

=

cvs

=

coversine

XW

=

vcs

=

vercosine

YS

=

cvc

=

covercosine

HR

=

hav

=

haversine

HU

=

hcv

=

hacoversine

HL

=

hvc

=

havercosine

HD

=

hcc

=

hacovercosine