Creating Trigonometric Functions by Wrapping the X-axis ...



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, [pic] (theta): sine [pic] = opposite/hypotenuse, cosine [pic] = adjacent/hypotenuse, and tangent [pic] = opposite/adjacent, where “opposite” refers to the side of the triangle that is opposite the acute angle [pic], “adjacent” refers to the shorter side of the triangle subtended by the angle [pic], and “hypotenuse” is the longest side of the right triangle (see figure 8.1).

[pic]

Figure 8.1: A right triangle with angle [pic]

While this approach to the trigonometric ratios is helpful for students who have only had “triangle geometry,” it has limitations when introducing students to the trigonometric functions. One major limitation is that the angle [pic] can only vary between 0 and 90º (or zero and π/2 radians). The trigonometric functions are functions of the angle [pic] (given in radians) and have an unlimited domain for the values of [pic]. Angle rotation around a circle can provide this unlimited domain. By convention, when starting with a unit circle on coordinate axes, the angle [pic] is measured as rotation anti-clockwise, starting from the positive x-axis (see figure 8.2).

[pic]

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 [pic] 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, [pic] ranges from zero to π/2. In figure 8.3, [pic] has been rotated into the second quadrant.

[pic]

Figure 8.3: The hypotenuse of the right triangle is now in Quadrant II.

For values of [pic] 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-[pic] or the supplement of [pic]. However, the adjacent side of the triangle now lies on the negative x-axis and consequently has a negative value for its length. Thus cosine [pic] in this second quadrant is negative, but sine [pic] is still positive as the opposite side is still in the positive half plane for the vertical axis. As the hypotenuse continues to rotate into the third quadrant, both adjacent and opposite sides will be negative and the referent angle inside the right triangle will be [pic]-π radians. 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-[pic]. As the rotation continues and the hypotenuse crosses back into the first quadrant, the value of [pic] is 2π plus the referent angle at vertex A (the original value of [pic] in figure 8.2). Thus, the whole cycle repeats itself every full rotation around the circle (multiples of 2π radians). It is important to realize that the length of the hypotenuse of the referent right triangle remains positive throughout this rotation (it is the 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 [pic] as the independent variable (x). Thus, the rotation of the angle [pic] has to be transformed into horizontal displacement along the x-axis.

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.

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).

[pic]

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-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).

[pic]

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.

[pic]

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?

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