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The trigonometric functions studied are defined geometrically rather than in terms of algebraic equations, but one of the goals of this course is to acquaint students with a more algebraic viewpoint toward these functions. Students should have a clear understanding that the definition of the trigonometric functions is made possible by the notion of similarity between triangles. A basic difficulty confronting students is one of superabundance: There are six trigonometric functions and seemingly an infinite number of identities relating to them. The situation is actually very simple, however. Sine and cosine are by far the most important of the six functions. Students must be thoroughly familiar with their basic properties, including their graphs and the fact that they give the coordinates of every point on the unit circle (Standard 2.0). Moreover, three identities stand out above all others: sin2 x + cos2 x = 1 and the addition formulas of sine and cosine: Trig 3.0 Students know the identity cos2(x) + sin2(x) = 1: 3.1. Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity). 3.2. Students prove other trigonometric identities and simplify others by using the identity cos2(x) + sin2(x) = 1. For example, students use this identity to prove that sec2(x) = tan2(x) + 1. Trig 10.0 Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities. Students should know the proofs of these addition formulas. An acceptable approach is to use the fact that the distance between two points on the unit circle depends only on the angle between them. Thus, suppose that angles a and b satisfy 0 < a < b, and let A and B be points on the unit circle making angles a and b with the positive x-axis. Then A = (cos a, sin a), B = (cos b, sin b), and the distance d(A, B) from A to B satisfies the equation: d(A, B)2 = (cos b " cos a)2 + (sin b " sin a)2. On the other hand, the angle from A to B is (b " a), so that the distance from the point C = (cos(b " a), sin(b " a)) to (1, 0) is also d(A, B) because the angle from C to (1, 0) is (b " a) as well. Thus: d(A, B)2 = (cos(b " a) " 1)2 + sin2(b " a). Equating the two gives the formula: cos(b " a) = cos a cos b + sin a sin b. From this formula both the sine and cosine addition formulas follow easily. Students should also know the special cases of these addition formulas in the form of half-angle and double-angle formulas of sine and cosine (Standard 11.0). These are important in advanced courses, such as calculus. Moreover, the addition formulas make possible the rewriting of trigonometric sums of the form A sin(x) + B cos(x) as C sin(x + D) for suitably chosen constants C and D, thereby showing that such a sum is basically a displaced sine function. This fact should be made known to students because it is important in the study of wave motions in physics and engineering. Students should have a moderate amount of practice in deriving trigonometric identities, but identity proving is no longer a central topic. Of the remaining four trigonometric functions, students should make a special effort to get to know tangent, its domain of definition EMBED Equation.DSMT4 , and its graph (Standard 5.0). The tangent function naturally arises because of the standard: Trig 7.0 Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line. Because trigonometric functions arose historically from computational needs in astronomy, their practical applications should be stressed (Standard 19.0). Among the most important are: Trig 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. Trig 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. These formulas have innumerable practical consequences. Complex numbers can be expressed in polar forms with the help of trigonometric functions (Standard 17.0). The geometric interpretations of the multiplication and division of complex numbers in terms of the angle and modulus should be emphasized, especially for complex numbers on the unit circle. Mention should be made of the connection between the nth roots of 1 and the vertices of a regular n-gon inscribed in the unit circle: Trig 18.0 Students know DeMoivre’s theorem and can give nth roots of a complex number given in polar form. Mathematical Analysis This discipline combines many of the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus and other advanced courses. It also brings a measure of closure to some topics first brought up in earlier courses, such as Algebra II. The functional viewpoint is emphasized in this course. Mathematical Induction The eight standards are fairly self-explanatory. However, some comments on four of them may be of value. The first is mathematical induction: MA 3.0 Students can give proofs of various formulas by using the technique of mathematical induction. This basic technique was barely hinted at in Algebra II; but at this level, to understand why the technique works, students should be able to use the technique fluently and to learn enough about the natural numbers. They should also see examples of why the step to get the induction started and the induction step itself are both necessary. Among the applications of the technique, students should be able to prove by induction the binomial theorem and the formulas for the sum of squares and cubes of the first n integers. Roots of Polynomials Roots of polynomials were not studied in depth in Algebra II, and the key theorem about them was not mentioned: MA 4.0 Students know the statement of, and can apply, the fundamental theorem of algebra. This theorem should not be proved here because the most natural proof requires mathematical techniques well beyond this level. However, there are “elementary” proofs that can be made accessible to some of the students. In a sense this theorem justifies the introduction of complex numbers. An application that should be mentioned and proved on the basis of the fundamental theorem of algebra is that for polynomials with real coefficients, complex roots come in conjugate pairs. Consequently, all polynomials with real coefficients can be written as the product of real quadratic polynomials. The quadratic formula should be reviewed from the standpoint of this theorem. Conic Sections The third area is conic sections (see Standard 5.0). Students learn not only the geometry of conic sections in detail (e.g., major and minor axes, asymptotes, and foci) but also the equivalence of the algebraic and geometric definitions (the latter refers to the definitions of the ellipse and hyperbola in terms of distances to the foci and the definition of the parabola in terms of distances to the focus and directrix). A knowledge of conic sections is important not only in mathematics but also in classical physics. Limits Finally, students are introduced to limits: MA 8.0 Students are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge. This standard is an introduction to calculus. The discussion should be intuitive and buttressed by much numerical data. The calculator is useful in helping students explore convergence and divergence and guess the limit of sequences. If desired, the precise definition of limit can be carefully explained; and students may even be made to memorize it, but it should not be emphasized. For example, students can be taught to prove why for linear functions EMBED Equation.DSMT4 , but it is more likely a ritual of manipulating µ s and ´ s in a special situation than a real understanding of the concept. The time can probably be better spent on other proofs (e.g., mathematical induction). REPRESENTATIVE PERFORMANCE OUTCOMES AND SKILLS In this course, students will: Know the identity EMBED Equation.DSMT4 Demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use these formulas to prove and/or simplify other trigonometric identities Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line Know the law of sines and the law of cosines and apply those laws to solve problems Determine the area of a triangle, given one angle and two adjacent sides Know DeMoivre’s theorem and can give nth roots of a complex number given in polar form Give proofs of various formulas by using the technique of mathematical induction Know the statement of, and can apply, the fundamental theorem of algebra Be familiar with the notion of the limit if a sequence and the limit of a function as the independent variable approaches a number or infinity. 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