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<p class="MsoNormal"><span style="font-size:11.0pt">Hello John,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Your ability to visualize and remember math is amazing.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">Regards<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Louis Maher<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Phone: 713-444-7838<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Email: ljmaher03@outlook.com<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"> Tactile-Talk <tactile-talk-bounces@nfbnet.org>
<b>On Behalf Of </b>John Miller via Tactile-Talk<br>
<b>Sent:</b> Friday, September 13, 2024 6:34 PM<br>
<b>To:</b> tactile-talk@nfbnet.org<br>
<b>Cc:</b> John Miller <johnmillerphd@hotmail.com><br>
<b>Subject:</b> [Tactile-Talk] Function Tangent<o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Hello,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I am excited to see all the individuals that have subscribed to tactile-talk. There are over one hundred members of the list at this time and I recognize many of you from prior
Zoom or Teams presentations and from on-point technical contributions to other NFBNET lists.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Thank you to David Andrews for making this list happen.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">As a blind person doing data visualization I find it helpful to memorize, memorize, memorize, and also to do repetition in support of that memorization.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I was reading the American Printing House for the Blind website
</span><u><span style="font-family:"Calibri",sans-serif;color:#467886"><a href="http://www.aph.org/" target="_blank" title="Protected by Outlook: http://www.aph.org/. Click or tap to follow the link."><span style="color:#467886">www.aph.org</span></a></span></u><span style="font-family:"Calibri",sans-serif;color:black"> about
the Monarch Tactile Graphic Display.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">One of the testimonials of a teacher for the blind wrote about a blind student found it so helpful to use the Monarch to view the graph of the trigonometric function tangent.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">It would be helpful if a service provider offered good quality tactile images on paper or thermoform with relevant braille labels showing the plots of sine, cosine, and tangent
on 3 separate pages.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I am quite familiar with the plots of sine and cosine. It might be a little bit harder for slightly curious blind individuals to visualize or generate a tactile plot for the function
tangent.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I want to remind myself about the plot of the function tangent and write up a description for all of you along the way.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I will use degrees going counter-clockwise from the positive x-axis where 360 degrees goes all the way around the unit circle.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The tangent of 0 is 0, the tangent of 30 degrees is 1/3 * sqrt(3) or approx. 0.577, the tangent of 45 degrees is 1, and the tangent of 60 degrees shoots up to sqrt(3) or approx.
1.732. As theta approaches 90 degrees from the left it goes to infinity.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Wikipedia states that the period of the tangent function is pi or 180 degrees. Wikipedia also states that tangent is an odd function. This means that f(-x) equals -f(x) or f(-theta)
equals -f(theta).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Recall that for theta, -30 degrees is the same as 330 degrees in radians and -n degrees is the same as 360 - n degrees.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Then for the negative angles:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The tangent of -30 degrees is -1/3 *sqrt(3) or approx. 0.577, the tangent of -45 degrees is -1,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">the tangent of 300 degrees is sqrt(3) or approx. 1.732 and as you approach -90 degrees from the right it is minus infinity.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I will now talk about common core math just for a little bit and then get back to talking about tangent.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">It is important to stay up to date with what is being taught in middle school and high school to help our young blind students.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">In the U.S. common core is taught in 42 states including the state of California where I live.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I dug a little bit deeper about common core.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I grew up in Nebraska getting my K12 education there and then moved to California. Four states never adopted common core. These include Texas, Nebraska, Virginia, and Alaska. If
you are helping a blind math student in one of these states perhaps math class might be similar to a math class you took in high school prior to 2010.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Common core was adopted in 2010 and may be evolving over time.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Common core covers a grab bag of units in a given year corresponding to a set of common core standards.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I attended a presentation at my sighted son's middle school about the California math curriculum.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The names of the high school math courses are integrated math 1, integrated math 2, integrated math 3, and integrated math 4.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">For the first two there is no honors option but there is an honors option for integrated math 3 and integrated math 4.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">There is no course taught called algebra and similarly for trigonometry in California.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">HumanWare on its website writes about how a blind student fell behind plotting quadratic functions using wiki sticks in algebra class.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The website said that using the Monarch would have been more effective for generating these plots.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Using wiki sticks is a good data visualization technique for the blind and using the Monarch may be faster but I would like to hear detailed reports about how students can use Monarch
to generate plots of quadratic functions that are deemed acceptable as deliverables for high school homework or exams.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I will say that the algebra class of the referenced student was not occurring in high school in the state of California.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">It is my guess that plotting of algebraic functions occurs in California in integrated math 2, usually in grade 10.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">In California there is no course in geometry.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">In California trigonometry gets covered in honors integrated math 3 usually in grade 11.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Integrated math 4 and honors integrated math 4 both cover calculus. I would make a guess that honors integrated math 4 can set the student up for getting a high score in AB AP
calculus.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">As an aside, my son has gone through a data science unit in grade 7 plotting and analyzing datasets on the number line.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The unit goes through demonstrating differences between mean and median and in some cases introduces some quite sophisticated concepts in statistics.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Now I will talk more about tangent. From a common core website I learned that sine, cosine, and tangent trigonometric functions are all covered as part of high school standards.
I am guessing they are usually taught in grade 11.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The common core website said you must know in what quadrants they are positive and negative. A web search for common core tangent positive provided some useful information.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The website introduces a pneumonic for the functions sine, cosine, and tangent and in which quadrant each of them is positive.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The pneumonic is A S T C for all students take calculus.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The "A" stands for in quadrant 1 all 3 functions are positive. The second letter "S" means that in quadrant 2 only sine is positive. The third letter "T" means that only tangent
is positive in quadrant 3, and the fourth letter "C" means that only cosine is positive in quadrant 4. The source said it is helpful to draw a unit circle which is a circle centered at the origin with radius 1.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Form a right triangle with one of the vertices at the origin and the angle at that vertex labeled theta.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">One of the edges of the triangle is the radius r drawn on a particular unit circle with angle theta. The edge from the origin to the end of the radius is the hypotenuse of a right
triangle from which you can derive the values of sine theta, cosine theta, and tangent theta.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The source talks about theta being measured counter-clockwise from the positive x-axis in radians. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I will use angles in degrees and point out that angles in radians equal angles in degrees divided by 180 and multiplied by pi.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Angles in radians read really well in the Nemeth braille code but angles in degrees read out just great when using JAWS or SIRI with speech output.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The hypotenuse of the right triangle is a radius of the unit circle and always has length one.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The other two sides of the right triangle are in the horizontal direction and in the vertical direction.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The right angle is between the horizontal edge and the vertical edge.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The horizontal edge is labeled adjacent since it is adjacent to the angle theta at the origin.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The vertical edge is labeled opposite since it is a nonadjacent edge to the angle theta.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Perhaps you can pause a moment and go sketch out the right triangle and the unit circle to help you visualize this write-up.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">For a triangle in quadrant 1 the horizontal segment is to the right along the positive x-axis.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The right angle is to the right of the origin.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The vertical segment starts at the right angle and goes up.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">For a triangle in quadrant 2 the horizontal segment goes to the left along the negative x-axis.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The right angle is to the left of the origin.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The vertical segment starts at the right angle and goes up.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">For a triangle in quadrant 3 the horizontal segment goes left along the negative x-axis.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The right angle is to the left of the origin.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The vertical segment starts at the right-angle and goes down.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">For a triangle in quadrant 4 the horizontal segment goes to the right along the positive x-axis.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The right angle is to the right of the origin.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The vertical segment starts at the right angle and goes down.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The definition for sine is the length of the opposite segment from theta divided by the length of the hypotenuse segment.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The definition for cosine is the length of the adjacent segment divided by the length of the hypotenuse.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The definition for tangent is the length of the opposite segment divided by the length of the adjacent segment.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">To review the top right quadrant is cartesian quadrant 1, the top left quadrant is cartesian quadrant 2, the bottom left quadrant is cartesian quadrant 3, and the bottom right quadrant
is cartesian quadrant 4.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">There is a very short video from
</span><u><span style="font-family:"Calibri",sans-serif;color:#467886"><a href="http://www.khanacademy.com/" target="_blank" title="Protected by Outlook: http://www.khanacademy.com/. Click or tap to follow the link."><span style="color:#467886">www.khanacademy.com</span></a></span></u><span style="font-family:"Calibri",sans-serif;color:black"> using
search words youtube khanacademy coordinate plane quadrant.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">It reminds us that the quadrants are labeled in roman numerals I, II, III, and IV.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Another video from khan academy says to help remember the quadrant numbers to place a print letter C (for Cartesian)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">centered at the origin. Now start tracing the letter C in the upper right corner at quadrant 1 and follow the c around counter-clockwise. As you trace the C you will move from
quadrants 1 to 2 to 3 and finally to 4.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">This memory device reminds us that there is a gap in the shape of the print letter C to the right of the origin.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">If you start at the top right and go around the C the only way you can progress is in the counter-clockwise direction.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">So now you will remember that quadrants are labeled in increasing value in the counter-clockwise direction starting with the top right quadrant.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Going back to the A S T C pneumonic tangent is positive in quadrants 1 and 3 but negative in quadrants 2 and 4.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Tangent is a ratio of the opposite and adjacent segments. It is positive if both elements of the ratio are positive like in quadrant 1 or when both elements of the ratio are negative
like in quadrant 3.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The denominator of tangent is negative in quadrant 2 while the numerator is positive so tangent is negative in quadrant 2.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The numerator of tangent is negative in quadrant 4 while the denominator is positive so tangent is negative in quadrant 4.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Time and again I go back to the following 16 row table when thinking about trigonometry<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">I will give it in terms of theta in degrees, x value, and y value but here x value means adjacent length and y value means opposite length.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">It just so happens that since the hypotenuse is of length 1 that x = cos(theta) = adjacent length and y = sin(theta) = opposite length.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The value of tangent is the third column value divided by the second column value.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">When the second column value is 0 and the third column is positive like for 90 degrees tangent is infinity.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">When the second column value is 0 and the third column value is negative like for 270 degrees tangent is minus infinity.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The lengths mentioned in some of the rows come from the 45-45-90 right triangle with hypotenuse 1.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">In that triangle the Pythagorean theorem gives x = sqrt(1-x^2) which reduces<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">to column 2 and 3 having values 1/2 * sqrt(2).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Some rows come from the 30-60-90 right triangle with hypotenuse 1 .<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Using the Pythagorean theorem if one edge is 1/2 then the other edge is sqrt(1-(1/2)^2) = sqrt(3/4) = 1/2 * sqrt(3).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The magnitude of x gets larger as you approach 0 and 180 degrees.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The magnitude of y gets larger as you approach 90 and 270 degrees.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">There are several kinds of symmetries that make memorizing the table easier.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The magnitude of x equals the magnitude of y for 45 degrees and 45 degrees plus n times 90.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">In quadrant 1 the value of x for 30 degrees is the same as the value of y for 60 degrees.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">The only thing left in this post is the table showing the value of cosine and sine for various angles around the unit circle.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">You can reference it when helping a blind student do trigonometry and skip the rest of the post otherwise.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">table with 16 rows:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">theta in degrees, x, y<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">0 degrees 1, 0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">30 degrees 1/2, 1/2 * sqrt(3)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">45 degrees 1/2 * sqrt(2), 1/2 * sqrt(2)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">60 degrees 1/2 * sqrt(3), 1/2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">90 degrees 0 1<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">120 degrees -1/2, 1/2 * sqrt(3)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">135 degrees -1/2 * sqrt(2), 1/2 * sqrt(2)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">150 degrees -1/2 * sqrt(3), 1/2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">180 degrees -1, 0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">210 degrees -1/2 * sqrt(3), -1/2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">225 degrees -1/2 * sqrt(2), -1/2 * sqrt(2)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">240 degrees -1/2, -1/2 * sqrt(3)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">270 degrees 0, -1<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">300 degrees 1/2 -1/2 * sqrt(3)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">325 degrees 1/2 * sqrt(2), -1/2 * sqrt(2)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">330 degrees 1/2 * sqrt(3), -1/2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">Very best,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Calibri",sans-serif;color:black">John Miller<o:p></o:p></span></p>
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