Texas Mathematics Teacher Fall/Winter 2019 - 15

Unit Circle Words Lesson
Lawrence Mark Lesser

Lesser's mathematical visual poem called "Unit Circle
Words", shown at right, originally appeared online on
March 15, 2019 (the day after Pi Day!) at
http://mathematicalpoetry.blogspot.com/2019/03/
unit-circle-word-by-lawrence-mark-lesser.html and is
reprinted with permission to accompany Lesser's
mini-lesson for it:
PREREQUISITES: Students should know (1) how to
name an angle with a one-letter or a three-letter name and
(2) how to define (in terms of triangle ratios or related to
a unit circle) the main six trigonometric functions: sine,
cosine, tangent, cotangent, secant, and cosecant.
QUESTIONS AND ACTIVITIES:
For questions 1-3, draw on the board the leftmost of the
three figures of the poem (see Figure 1), omitting the
words "SING" and "TANG." Tell students that circle G is
a unit circle.
1. Find the length of the shorter vertical segment in
terms of one of the main six trigonometric functions
of ∠G, using the usual three-letter abbreviation for
that trig function. What do you notice?
2. Find the length of the longer vertical segment in
terms of one of the main trigonometric functions of
∠G, again using the usual three-letter abbreviation
for that trig function so that a four-letter word is
formed.
3. Change the letter G to a different letter so that a
trigonometric function of the length of the side
adjacent to the central angle in the smaller right
triangle will be a word.
For activities 4-6, draw the upper right of the three figures
of the poem, but omit the words "TANGLE", SINGLE",
and "COSILY." Tell students that circle L is a unit circle.
4. In the smaller right triangle, find the length of the
side adjacent to the central angle using the usual
three-letter abbreviation for that trig function and
a three-letter name for the angle so that a six-letter
word is formed.
5. In the larger right triangle, find the length of the side
opposite the central angle using the usual
three-letter abbreviation for that trig function and a
three-letter name for the angle so that a six-letter
word is formed.
6. In the smaller right triangle, find the length of the
side opposite the central angle using the usual
three-letter abbreviation for that trig function and
a three-letter name for the angle so that a six-letter
word is formed.

Figure 1
For question 7, draw the same type of figure for the larger
right triangle where the circle's center is N, the point of
tangency is A, and the other vertex of the larger right
triangle is T.
7. Find the length NT, using the usual three-letter
abbreviation for a trig function and a three-letter
name for the angle so that a six-letter word in
English is formed.
Is there another answer for the length in question #7
for a different triple of letters?
To support or reinforce the lesson, students can be given
the Figure 2 unit circle diagram from Lesser's letter on p.
148 of the October 2004 Mathematics Teacher and asked to
find segments that represent the lengths of each of the six
major trigonometry functions.
Answers can be found on page 22

Activities similar in structure to problems 4-6 can also be
applied to Figure 1.

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Fall/Winter 2019 | 15
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Texas Mathematics Teacher Fall/Winter 2019

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