In grades 9-12, the mathematics curriculum should include the study of trigonometry so that all students can--

• apply trigonometry to problem situations involving triangles;
• explore periodic real-world phenomena using the sine and cosine functions;

and so that, in addition, college-intending students can--

• understand the connection between trigonometric and circular functions;
• use circular functions to model periodic real-world phenomena;
• apply general graphing techniques to trigonometric functions;
• solve trigonometric equations and verify trigonometric identities;
• understand the connections between trigonometric functions and polar coordinates, complex numbers, and series.

Focus

Trigonometry has its origins in the study of triangle measurement. Many real-world problems, including those from the fields of navigation and surveying, require the solution of triangles. In addition, important mathematical topics, such as matrix representations of rotations, direction angles of vectors, polar coordinates, and trigonometric representations of complex numbers, require trigonometric ratios, further underscoring the connections between geometry and algebra.

Natural generalizations of the ratios of right-angle trigonometry give rise to both trigonometric and circular functions. These functions, especially the sine and cosine, are mathematical models for many periodic real-world phenomena, such as uniform circular motion, temperature changes, biorhythms, sound waves, and tide variations. Although all students should explore data from such phenomena, college-intending students should identify and analyze the corresponding trigonometric models. These students should also study identities involving trigonometric expressions and inverses of trigonometric functions, together with their applications to the solution of equations and inequalities.

Scientific calculators can and should significantly facilitate the teaching of trigonometry, providing more class time and computational power to develop conceptual understanding and address realistic applications. Graphing utilities provide dynamic tools that permit students to model many realistic problem situations using trigonometric equations or inequalities. Consistent with the other standards, graphing utilities also should play an important role in students' development of an understanding of the properties of trigonometric functions and their inverses. In addition, college-intending students should solve trigonometric equations and inequalities by computer-based methods, such as those described in the standard on algebra.

Discussion

All students should apply trigonometric methods to practical situations involving triangles. As an example, consider a right-triangle surveying problem with which cartographers are frequently confronted.

Determine the angle of depression between two markers on a contour map with different elevations.

Students would first develop a geometric model (fig. 9.1) based on information read from the map. They would then identify a trigonometric ratio appropriate to the situation, write the corresponding equation, use a calculator to readily obtain a numerical answer, and then interpret this value to the appropriate degree of accuracy in terms of the given units of measure. College-intending students also should derive and apply the laws of sines and cosines to problem situations involving general triangles.

Fig. 9.1. A practical application of trigonometry

All students should use the sine and cosine functions to model periodic real-world phenomena. One setting with which the majority of students are familiar is that of a Ferris wheel.

Suppose a Ferris wheel with a radius of 25 feet makes a complete revolution in 12 seconds. Develop a mathematical model that describes the relationship between the height h of a rider above the bottom of the Ferris wheel (4 feet above the ground) and time t.

This problem can be addressed within the core curriculum by students at several possible levels of formalism.

Level 1: At this level, students would first develop a table of t- and h-values. Assuming that the rider is at the bottom of the Ferris wheel when t = 0, students can easily determine values of h for t = 0, 3, 6, 9, 12. For t-values between these numbers, values of h could be estimated from a scale drawing of the Ferris wheel as in figure 9.2(a). By plotting the collected data (fig. 9.2(b)) and noting the periodicity of the function, students may conjecture that the graph has a sinusoidal shape and thereby predict its shape for larger values of t.

Fig. 9.2. Modeling the position of a rider on a Ferris wheel

Level 2: Students at this level would be given an equation for h(t) (such as h(t) = - 25 cos (/6)t + 25) and asked to graph it and then to analyze their graphs. The interpretation of their graphs should focus on the contextual meaning of the local maximum and minimum points, finding h-values for given t-values and t-values for given h-values, and finding the number of revolutions for some (large) t-value and the time t required for a given number of revolutions. Finally, students would explore the changes in the graph for a Ferris wheel that has a different radius or rate of revolution.

Level 3: Recognizing that the graph obtained through experiences such as those in level 1 is that of a function of the form h(t) = a cos (bt) + c, students at this level would proceed to determine a, b, and c by comparing the graph of f (t) = cos t to their graph. This analysis would suggest the need to reflect the graph of f across the t-axis and then to adjust the amplitude, period, and shift in the vertical direction.

Level 4: At this level, students would use right-triangle trigonometry and simple proportions (see fig. 9.3) to derive the parametric representation of a point P = (x(t), y(t)) on the rotating Ferris wheel as a function of time, thereby establishing that the height is a sinusoidal function of t. They could then use a parametric graphing utility to simulate the motion of a point moving on the Ferris wheel.

Fig. 9.3. Parametric representation of P

We again emphasize that the entry and exit levels with respect to the treatment of this, or any, particular topic is largely determined by the background of the students and their performance in the activity itself. For some students these two levels may be the same. Seldom would any student progress through all levels within a single unit of study.

Concepts related to trigonometric functions such as amplitude, period, and phase shift should be introduced to college-intending students through real-world applications. These students will have had experience with graphs of functions of the form y = af(bx + c) + d, including the investigation of the effects of changing the parameters a, b, c, and d on the graph of y = f(x). Thus, after appropriate computer-graphing experiences, they should be able to sketch quickly, without the aid of a computer, the graph of a function like y = 3 sin (x + 2) by applying two transformations to the graph of y = sin x.

College-intending students also should have opportunities to verify basic trigonometric identities, such as sec²(A) = 1 + tan²(A), since this activity improves their understanding of trigonometric properties and provides a new setting for deductive proof. Only minimal amounts of class time should be devoted to verifying identities, however, and artificially complicated identities, such as csc⁶(x) - cot⁶(x) = 1 + 3 csc²(x) cot ²(x), should be avoided altogether.

College-intending students should also develop an understanding of the connections between trigonometric functions and the topics of polar coordinates, complex numbers, and series. Using a calculator or a computer, for example, students can investigate the power-series expansion of the sine function numerically and graphically. Figure 9.4 illustrates how the first five terms of the series expansion for the sine function very closely approximate the values of the sine function for |x| 4.

Fig. 9.4. Series expansion of the sine function

Students could use a graphing utility to explore such issues as the number of terms of the expansion necessary for the series to closely approximate the sine function for |x| 10. Such an approach can lead to valuable discussions of limits and errors in approximations.

Trigonometry not only remains an important and powerful tool for science and engineering but also continues to provide an aesthetic attraction for many students through its regularities and symmetries. Calculator and computer technology makes both aspects of the subject readily accessible to a wider range of students and at an earlier age level. This in turn provides opportunities for greater integration of trigonometry with geometry and algebra.