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 the triangle subtended by the angle longest side of the right triangle see figure 8 1 and hypotenuse is the Figure 8 1 A right triangle with angle 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 can only vary between 0 and 90 or zero and 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 domain By convention when starting with a unit circle on coordinate axes the angle is measured as rotation anti clockwise starting from the positive x axis see figure 8 2 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 to 2 In figure 8 3 has been rotated into the second quadrant ranges from zero Olive Oppong Transforming Mathematics with GSP 4 page 99 Figure 8 3 The hypotenuse of the right triangle is now in Quadrant II For values of between 2 and 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 or the supplement of 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 in this second quadrant is negative but sine 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 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 2 As the rotation continues and the hypotenuse crosses back into the first quadrant the value of is 2 plus the referent angle at vertex A the original value of in figure 8 2 Thus the whole cycle repeats itself every full rotation around the circle multiples of 2 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 as the independent variable x Thus the rotation of the angle 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 Olive Oppong Transforming Mathematics with GSP 4 page 100 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 Olive Oppong Transforming Mathematics with GSP 4 page 101 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 xcoordinate 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