Natsoulas, Anthula

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considered, each faculty member may feel unhappy with the new salary. The reader is encouraged to study actual budget data to see how frequently this apparent anomaly noted in Tables I and II actually occurs, MAKING TRIGONOMETRIC RATIOS MEANINGFUL Anthula Natsoulas University of Toledo Toledo, Ohio High school students working with trigonometric ratios often do not have a complete understanding of the meaning of the functions, not relating them to similar right triangles and the resulting equality of ratios of the corresponding sides. One way to make the trigonometric functions more "real" to students and to integrate use of both the calculator and geometric construc-tions is to have students construct similar right triangles, take measurements, calculate ratios and compare them. The results, presented in tabular form,. make the trigonometric functions appear to be a logical consequence. activities is outlined below. A suggested sequence of Begin by giving students two activity sheets each contain-ing four line segments of different lengths and orientations. The line segments on one page may all be labeled AC and those on the second A'C' Instruct students to construct right triangles ABC and A'·B'·C' for each of the given line segments for specified degree measures of angles A and A (where the measure of angle A 1s equal to the measure of angle A~) and with the right angle at C. A protractor could be used to draw angles A and A' or, if their degree measures are carefully chosen, angles A and A' may -lJ-be constructed with s~raight edge and compass as should the right angle C (see Figures 1 and 2). Once the triangles are completed, students are asked to measure the length of legs AC and BC and hypotenuse AB and legs A'·c', B'·c•, and hypotenuse A'·B'. Provide students with two tally sheets, one for each set of right triangles on which they can record their results and calculate the indicated ratios (see Figure 3). A sample of a completed table is shown in Figure 4. Questioning may be used to direct students I attention to the relationships which the completed tables demonstrate, focusing in on the calculated ratios across all right triangles with the same acute angle. A discussion may follow regarding the relationship among the four right triangles of each set and the patterns which appear to emerge in the tables. Comparing the results of each set of right triangles, within each set, the tables clearly show the three calculated ratios to be very nearly equal for those triangles with the same acute angle irrespective of the lengths of the sides. Students may further be encouraged to verbalize the reason for such a relationship, relating back to similar triangles and the equality of corresponding ratios. A formal proof may be presented at this point. Following such a sequence of activities, students would readily agree that it seems reasonable to apply specific names to such ratios and thus the terms sine, cosine, and tangent may be introduced and formal definitions of these functions can be made. This approach is totally appropriate for a tenth grade geometry class which is seeing formal definitions of trigonomet-ric functions for the first time and can have more meaning than the simple statement of such ratios. Having the advantage of using a calculator should convince students that the ratios will come out the same regardless of the lengths of the sides of the triangles or the measures of the angles (let them choose their own lengths and angle measure). Furthermore, the activity allows integration of constructions and application of the problem solving heuristic of looking for a pattern. -14-,:) ,.;; Triangles Set I Directions: For each of the four line segments given below, construct a right angle at C. Co~plete right triangle ABC such that the measure of angle A= 40. A C A C / Figure 1 Triangle Set II Directions: For each of the four line segments given below, construct a right angle at C, Co~plete right triangle ABC such that the measure of angle A= 25. A C. ~ A C. ~ Figure 2 -15-be constructed with st~aight edge and compass as should the right angle C (see Figures 1 and 2). Once the triangles are completed, students are asked to measure the length of legs AC and BC and hypotenuse AB and legs A•c•, B•c•, and hypotenuse A·B'. Provide students with two tally sheets, one for each set of right triangles on which they can record their results and calculate the indicated ratios (see Figure 3). A sample of a completed table is shown in Figure 4. Questioning may be used to direct students' attention to the relationships which the completed tables demonstrate, focusing in on the calculated ratios across all right triangles with the same acute angle. A discussion may follow regarding the relationship among the four right triangles of each set and the pat terns which appear to emerge in the tables. Comparing the results of each set of right triangles, within each set, the tables clearly show the three calculated ratios to be very nearly equal for those triangles with the same acute angle irrespective of the lengths of the sides. Students may further be encouraged to verbalize the reason for such a relationship, relating back to similar triangles and the equality of corresponding ratios. A formal proof may be presented at this point. Following such a sequence of activities, students would readily agree that it seems reasonable to apply specific names to such ratios and thus the terms sine, cosine, and tangent may be introduced and formal definitions of these functions can be made. This approach is totally appropriate for a tenth grade geometry class which is seeing formal definitions of trigonomet-ric functions for the first time and can have more meaning than the simple statement of such ratios. Having the advantage of using a calculator should convince students that the ratios will come out the same regardless of the lengths of the sides of the triangles or the measures of the angles (let them choose their own lengths and angle measure). Furthermore, the activity allows integration of constructions and application of the problem solving heuristic of looking for a pattern. -14-I Triangles Set I Directions: For each of the four line segments given below, construct a right angle at C. Co~plete right triangle ABC such that the measure of angle A= 40. A C A C / Figure l Triangle Set II Directions: For each of the four line segments given below, construct a right angle at C, Co~plete right triangle ABC such that the measure of angle A= 25. A C. A c.. ~ Figure 2 -15-Summary Table:Right Triangle Measurements Triangle Triangle Triangle Triangle I II III IV measure of angle A length of leg AC length of leg BC length of hypotenuse Al: ratio oT AC to AB ratio of BC to AB ratio of AC to BC !) Figure J s Tabl -- Right T--' le M ............... __,~ --- - -- t --------Triangle Triangle Triangle Triangle I II III IV measure ot· 40° 40° 40° 40° angle A length of 7.8 5.3 6.4 J.9 leg AC length of 6.2 4.4 5,4 J.2 leg BC 1.ength of 10.J 6.9 8.2 5.1 hypotenuse AI ratio or ,76 ,77 ,78 ,76 AC to AB ratio or .60 .64 .66 .6J BC to AB ratio o:i: 1.26 1.20 1.19 1.22 AC to BC Figure 4 i111i -16-,11: ·~- ~ .. "---~-----···-----If\ f!J !r r .! WHY CERTAIN GEOMETRIC RATIOS ARE CALLED "HARMONIC" Kenneth Cummins Kent State University Kent, Ohio In school geometry there are various instances of the ratio of interior division ratio to ex-terior division ratio--or cross-ratio. One of these arises in the study of the bisectors of the in-terior and exterior angles at a vertex of a triangle, It can be shown that if c't and ·c? are interior £ and exterior angle bisectors at C then AD= CA - - AE = CA and -EB CB whence DB CB '" AD DB -AE EB D 8 Figure 1 or that AD:DB = -AE:EB which says that the interior division ratio of AB formed by D equals the negative of the exterior division ratio of AB formed by E, or that the ratio of the two ratios (internal and external) above equals -1. Another case arises when from an external point E two tangents are drawn to a circle as shown in the accompanying figure. Regarding points D and E as in- A terior and exterior points o\ division of segment A~, it can be shown by high school methods that AD DB -AE EB or that AD I DB AE EB = -1 Figure 2 Although not traditionally found in high school geometry, one can prove by using the Theorems of Menelaus and Ceva that in this com-plete quadrilateral points D and E separate the segment AB such that the interior ratio/exterior ratio ~~ l ------~ f,.. l) B E--1. Figure 3 -17-

Making Trigonometric Ratios Meaningful

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Making Trigonometric Ratios Meaningful

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