[Tactile-Talk] Function Tangent

Fri Sep 13 23:34:17 UTC 2024

Hello,
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.
Thank you to David Andrews for making this list happen.

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.

I was reading the American Printing House for the Blind website www.aph.org<http://www.aph.org/> about the Monarch Tactile Graphic Display.
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.
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.
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.
I want to remind myself about the plot of the function tangent and write up a description for all of you along the way.
I will use degrees going counter-clockwise from the positive x-axis where 360 degrees goes all the way around the unit circle.
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.
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).
Recall that for theta, -30 degrees is the same as 330 degrees in radians and -n degrees is the same as 360 - n degrees.
Then for the negative angles:
The tangent of -30 degrees is -1/3 *sqrt(3) or approx. 0.577, the tangent of -45 degrees is -1,
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.
I will now talk about common core math just for a little bit and then get back to talking about tangent.

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.
In the U.S. common core is taught in 42 states including the state of California where I live.
I dug a little bit deeper about common core.
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.
Common core was adopted in 2010 and may be evolving over time.
Common core covers a grab bag of units in a given year corresponding to a set of common core standards.
I attended a presentation at my sighted son's middle school about the California math curriculum.
The names of the high school math courses are integrated math 1, integrated math 2, integrated math 3, and integrated math 4.
For the first two there is no honors option but there is an honors option for integrated math 3 and integrated math 4.
There is no course taught called algebra and similarly for trigonometry in California.
HumanWare on its website writes about how a blind student fell behind plotting quadratic functions using wiki sticks in algebra class.
The website said that using the Monarch would have been more effective for generating these plots.
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.
I will say that the algebra class of the referenced student was not occurring in high school in the state of California.
It is my guess that plotting of algebraic functions occurs in California in integrated math 2, usually in grade 10.
In California there is no course in geometry.
In California trigonometry gets covered in honors integrated math 3 usually in grade 11.
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.
As an aside, my son has gone through a data science unit in grade 7 plotting and analyzing datasets on the number line.
The unit goes through demonstrating differences between mean and median and in some cases introduces some quite sophisticated concepts in statistics.

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.
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.
The website introduces a pneumonic for the functions sine, cosine, and tangent and in which quadrant each of them is positive.
The pneumonic is A S T C for all students take calculus.
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.
Form a right triangle with one of the vertices at the origin and the angle at that vertex labeled theta.
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.
The source talks about theta being measured counter-clockwise from the positive x-axis in radians.
I will use angles in degrees and point out that angles in radians equal angles in degrees divided by 180 and multiplied by pi.
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.
The hypotenuse of the right triangle is a radius of the unit circle and always has length one.
The other two sides of the right triangle are in the horizontal direction and in the vertical direction.
The right angle is between the horizontal edge and the vertical edge.
The horizontal edge is labeled adjacent since it is adjacent to the angle theta at the origin.
The vertical edge is labeled opposite since it is a nonadjacent edge to the angle theta.
Perhaps you can pause a moment and go sketch out the right triangle and the unit circle to help you visualize this write-up.

For a triangle in quadrant 1 the horizontal segment is to the right along the positive x-axis.
The right angle is to the right of the origin.
The vertical segment starts at the right angle and goes up.
For a triangle in quadrant 2 the horizontal segment goes to the left along the negative x-axis.
The right angle is to the left of the origin.
The vertical segment starts at the right angle and goes up.
For a triangle in quadrant 3 the horizontal segment goes left along the negative x-axis.
The right angle is to the left of the origin.
The vertical segment starts at the right-angle and goes down.

For a triangle in quadrant 4 the horizontal segment goes to the right along the positive x-axis.
The right angle is to the right of the origin.
The vertical segment starts at the right angle and goes down.

The definition for sine is the length of the opposite segment from theta divided by the length of the hypotenuse segment.
The definition for cosine is the length of the adjacent segment divided by the length of the hypotenuse.
The definition for tangent is the length of the opposite segment divided by the length of the adjacent segment.
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.
There is a very short video from www.khanacademy.com<http://www.khanacademy.com/> using search  words youtube khanacademy coordinate plane quadrant.
It reminds us that the quadrants are labeled in roman numerals I, II, III, and IV.
Another video from khan academy says to help remember the quadrant numbers to place a print letter C (for Cartesian)
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.
This memory device reminds us that there is a gap in the shape of the print letter C to the right of the origin.
If you start at the top right and go around the C the only way you can progress is in the counter-clockwise direction.
So now you will remember that quadrants are labeled in increasing value in the counter-clockwise direction starting with the top right quadrant.

Going back to the A S T C pneumonic tangent is positive in quadrants 1 and 3 but negative in quadrants 2 and 4.
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.
The denominator of tangent is negative in quadrant 2 while the numerator is positive so tangent is negative in quadrant 2.
The numerator of tangent is negative in quadrant 4 while the denominator is positive so tangent is negative in quadrant 4.

Time and again I go back to the following 16 row table when thinking about trigonometry
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.
It just so happens that since the hypotenuse is of length 1 that x = cos(theta) = adjacent length and y = sin(theta) = opposite length.
The value of tangent is the third column value divided by the second column value.
When the second column value is 0 and the third column is positive like for 90 degrees tangent is infinity.
When the second column value is 0 and the third column value is negative like for 270 degrees tangent is minus infinity.
The lengths mentioned in some of the rows come from the 45-45-90 right triangle with hypotenuse 1.
In that triangle the Pythagorean theorem gives x = sqrt(1-x^2) which reduces
to column 2 and 3 having values 1/2 * sqrt(2).
Some rows come from the 30-60-90 right triangle with hypotenuse 1 .
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).
The magnitude of x gets larger as you approach 0 and 180 degrees.
The magnitude of y gets larger as you approach 90 and 270 degrees.
There are several kinds of symmetries that make memorizing the table easier.
The magnitude of x equals the magnitude of y for 45 degrees and 45 degrees plus n times 90.
In quadrant 1 the value of x for 30 degrees is the same as the value of y for 60 degrees.

The only thing left in this post is the table showing the value of cosine and sine for various angles around the unit circle.
You can reference it when helping a blind student do trigonometry and skip the rest of the post otherwise.

table with 16 rows:
theta in degrees, x, y
0 degrees 1, 0
30 degrees 1/2, 1/2 * sqrt(3)
45 degrees 1/2 * sqrt(2), 1/2 * sqrt(2)
60 degrees 1/2 * sqrt(3), 1/2
90 degrees 0 1
120 degrees -1/2, 1/2 * sqrt(3)
135 degrees -1/2 * sqrt(2), 1/2 * sqrt(2)
150 degrees -1/2 * sqrt(3), 1/2
180 degrees -1, 0
210 degrees -1/2 * sqrt(3), -1/2
225 degrees -1/2 * sqrt(2), -1/2 * sqrt(2)
240 degrees -1/2, -1/2 * sqrt(3)
270 degrees 0, -1
300 degrees 1/2 -1/2 * sqrt(3)
325 degrees 1/2 * sqrt(2), -1/2 * sqrt(2)
330 degrees 1/2 * sqrt(3), -1/2

Very best,
John Miller
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://nfbnet.org/pipermail/tactile-talk_nfbnet.org/attachments/20240913/289b9f46/attachment.htm>