Olive & Oppong: Transforming Mathematics with GSP 4, page 97
Chapter 8: Trigonometry of the unit circle with GSP
Most high-school curricula introduce the trigonometric ratios using ratios in a right triangle. The
three main ratios are usually defined for an acute angle,  (theta): sine  = opposite/hypotenuse,
cosine  = adjacent/hypotenuse, and tangent  = opposite/adjacent, where “opposite” refers to
the side of the triangle that is opposite the acute angle  , “adjacent” refers to the shorter side of
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the triangle subtended by the angle  , and “hypotenuse” is the longest side of the right triangle
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(see figure 8.1).
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Figure 8.1: A right triangle with angle 
While this approach to the trigonometric ratios is helpful for students who have only had
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“triangle geometry,” it has limitations when introducing students to the trigonometric functions.
One major limitation is that the angle  can only vary between 0 and 90º (or zero and π/2
radians). The trigonometric functions are functions of the angle  (given in radians) and have an
unlimited domain for the values of  . Angle rotation around a circle can provide this unlimited
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domain. By convention, when starting with a unit circle on coordinate axes, the angle  is
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measured as rotation anti-clockwise, starting from the positive x-axis (see figure 8.2).
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Olive & Oppong: Transforming Mathematics with GSP 4, page 98
Figure 8.2: Trigonometric right triangle in the Unit Circle.
A major advantage to using the UNIT circle for this construction is that the hypotenuse now has
length one unit, and therefore the sine and cosine ratios of the angle  have the values of the
lengths of the opposite and adjacent sides, respectively. In figure 8.2 above, the right triangle is
shown in the first quadrant of the coordinate system. Within this quadrant,  ranges from zero
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to π/2. In figure 8.3,  has been rotated into the second quadrant.
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Figure 8.3: The hypotenuse of the right triangle is now in Quadrant II.
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For values of  between π/2 and π radians, the trigonometric right triangle has shifted into the
second quadrant of the coordinate system and the referent angle for the trig ratios is now π-  or
the supplement of  . However, the adjacent side of the triangle now lies on the negative x-axis
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and consequently has a negative value for its length. Thus cosine  in this second quadrant is
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negative, but sine  is still positive as the opposite side is still in the positive half plane for the
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vertical axis. As the hypotenuse continues to rotate into the third quadrant, both adjacent and
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opposite sides will be negative and the referent angle inside the right triangle will be  -π radians.
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Continuing the rotation into quadrant IV, gives a positive adjacent side and a negative opposite
side and the referent angle at vertex A is now 2π-  . As the rotation continues and the
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hypotenuse crosses back into the first quadrant, the value of  is 2π plus the referent angle at
vertex A (the original value of  in figure 8.2). Thus, the whole cycle repeats itself every full
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rotation around the circle (multiples of 2π radians). It is important to realize that the length of
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the hypotenuse of the referent right triangle remains positive throughout this rotation (it is the
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radius of the unit circle). The familiar graphs of the trigonometric functions are generated by
plotting the value of the trigonometric ratio (y) for any value of the angle  as the independent
variable (x). Thus, the rotation of the angle  has to be transformed into horizontal displacement
along the x-axis.
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Most GSP sketches that generate trigonometric functions through animation of points on the unit
circle and the x-axis are limited to one revolution of the point about the circle. The construction
is, in effect, unwrapping the circumference of the circle along the x-axis (this can be done in a
variety of ways). In the following activity we demonstrate an alternative approach that can be
viewed as wrapping the x-axis around the unit circle. This alternative approach has the
advantage of providing a more extensive domain and range for the trigonometric functions.
Activity 8.1: Construction Steps for the Sine Curve
1. On a new sketch create a set of axes using the Graph menu item.
2. Use the origin and unit point to create a unit circle.
3. Place a free point on the x-axis. Rename this point "x".
4. Set your angle unit to Radians using the Preferences under the Edit menu.
5. Measure the x-coordinate of point x.
Olive & Oppong: Transforming Mathematics with GSP 4, page 100
6. Multiply the x-coordinate by 1 radian using the calculator.
7. Select the origin point as a center of rotation (double click).
8. Rotate the unit point (point B) by the measurement from step 6 (x radians).
At this stage in the construction you should have a point B' on the unit circle. The arc BB'
(counter clockwise around the circle) has the same length as the segment from the origin to the
point x (why?). The rotation of B to B' has, in effect, wrapped that portion of the x-axis around
the circle. Move point x back and forth along the x-axis to see the effect on B'. The rest of the
steps in the construction of the sine curve simply connect the motions of the input variable (x)
with the output variable B'.
9. Construct a line through point x perpendicular to the x-axis.
10. Construct a line through point B' parallel to the x-axis.
11. Construct the intersection point of these two lines. Rename this point "sin x".
12. Select point sin x and point x (in that order) and choose Locus from the Construct menu.
The graph of the sine function should appear as in figure 8.4 below. You can increase the
visible range and smoothness of the function by increasing the number of objects in your
locus (advanced preferences). Moving point x along the x-axis will also change the range of
the function (the locus follows point x).
Figure 8.4: Graph of Sine x from GSP
The reason that this construction produces the sine function (rather than any of the other
trigonometric functions) is that the height of point B' above the x-axis is equal in magnitude and
sign (positive or negative) to the sine of x (it is the “opposite” side in the referent right triangle as
seen in figure 8.2 above). Thus the y-coordinate of point "sin x" is, in fact, the sine of the x-
Olive & Oppong: Transforming Mathematics with GSP 4, page 101
coordinate of point x. In figure 8.5, I have constructed the triangle formed by the origin (A),
point B' and the perpendicular from B' to the x-axis (point E).
Figure 8.5: The trig triangle in the unit circle.
All of the trigonometric ratios can be formed from this triangle (see figure 8.1 above). Because
the radius of the unit circle is one unit, the distance EB' gives the value of the sine of angle EAB'
(and of angle BAB'). It is possible to construct appropriate triangles on this unit circle,
controlled by the point B', that provide segments perpendicular to the x-axis whose distance
above (or below) the x-axis is the same value as each of the other trigonometric ratios. Using the
vertical segments in these special triangles, lines parallel to the x-axis can be constructed that
intersect the vertical line through point x to produce the locus points for each of the six
trigonometric functions. I leave these constructions as a challenge to the reader! Figure 8.6
illustrates one possibility for constructing tan x.
Assignment 8.2: Construct a triangle based on the standard right triangle shown in figures 8.2
and 8.5 that provides a vertical segment equal in length to the adjacent side of the referent circle.
Use this vertical segment to construct the locus point for cosine x.
Assignment 8.3: Construct a triangle (see figure 8.6) that will provide a vertical segment that will
have length equivalent to the tangent of x radians. Use this triangle to construct the locus point
for tan x.
Olive & Oppong: Transforming Mathematics with GSP 4, page 102
Figure 8.6: Graphs of Sin x and of Tan x using the unit circle in GSP
Assignment 8.4 (Extra Challenge): Construct triangles based on the referent triangle that provide
vertical segments equivalent in length to the values of the other three trigonometric functions:
Secant, cosecant and cotangent of x.
Reflections: We started this chapter with a review of the right triangle trigonometry and then
connected this typical approach for generating the trigonometric ratios to the rotation of a point
around a unit circle. Compare these two approaches to introducing students to the trigonometric
ratios and subsequently to the trigonometric functions. What are the pros and cons of each
approach? Would it be easier for students to start with rotation around a unit circle rather than
with a right triangle? Why or why not?
In what ways did the dynamic motion of points along the x-axis and the rotating points
around the circle help you (or not) to understand the shapes of the graphs of the trigonometric
functions? Do the dynamic possibilities with GSP help (or not) to visualize the periodic nature of
the trigonometric functions?