  The California Mathematics Content Standards

Trigonometry Mathematics Content Standards

Trigonometry uses the techniques that students have previously learned from the study
of algebra and geometry . The trigonometric functions studied are defined geometrically
rather than in terms of algebraic equations. Facility with these functions as well as the
ability to prove basic identities regarding them is especially important for students
intending to study calculus, more advanced mathematics, physics and other sciences,
and engineering in college.

 1.0 Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians. Note: The sample problems illustrate the standards and are written to help clarify them. Some problems are written in a form that can be used directly with students; others will need to be modified before they are used with students. 2.0 Students know the definition of sine and cosine as y- and x- coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions. Find an angle β between 0 and 2π such that cos (β) = cos (6π/7) and sin (β) = -sin (6π/7). Find an angle θ between 0 and 2π such that sin (θ) = cos (6π/7) and cos (θ) = sin (6π/7). 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. 4.0 Students graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. On a graphing calculator, graph the function f(x) = sin (x) cos (x). Select a window so that you can carefully examine the graph. 1. What is the apparent period of this function? 2. What is the apparent amplitude of this function? 3. Use this information to express f as a simpler trigonometric function. 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. 6.0 Students know the definitions of the secant and cosecant functions and can graph them. 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. 8.0 Students know the definitions of the inverse trigonometric functions and can graph the functions. 9.0 Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. 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. Use the addition formula for sine to find an expression for sin (75°). 11.0 Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities. 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. A vertical pole sits between two points that are 60 feet apart. Guy wires to the top of that pole are staked at the two points. The guy wires are 40 feet and 35 feet long. How tall is the pole? 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. 15.0 Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa. 16.0 Students represent equations given in rectangular coordinates in terms of polar coordinates. Express the circle of radius 2 centered at (2, 0) in polar coordinates. 17.0 Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form. 18.0 Students know DeMoivre’s theorem and can give nth roots of a complex number given in polar form. 19.0 Students are adept at using trigonometry in a variety of applications and word problems. A lighthouse stands 100 feet above the surface of the ocean. From what distance can it be seen? (Assume that the radius of the earth is 3,960 miles.)

Mathematical Analysis Mathematics Content Standards

This discipline combines many of the trigonometric, geometric, and algebraic techniques
needed to prepare students for the study of calculus and strengthens their conceptual
understanding of problems and mathematical reasoning in solving problems. These
standards take a functional point of view toward those topics. The most significant new
concept is that of limits. Mathematical analysis is often combined with a course in trigonometry
or perhaps with one in linear algebra to make a yearlong precalculus course.

 Note: The sample problems illustrate the standards and are written to help clarify them. Some problems are written in a form that can be used directly with students; others will need to be modified before they are used with students. 1.0 Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically. 2.0 Students are adept at the arithmetic of complex numbers. They can use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. They know the proof of DeMoivre’s theorem. 3.0 Students can give proofs of various formulas by using the technique of mathematical induction. Use mathematical induction to show that the sum of the interior angles in a convex polygon with n sides is ( n - 2)×180°. 4.0 Students know the statement of, and can apply, the fundamental theorem of algebra. Find all cubic functions of x that have zeros at x = -1 and x = 2 and nowhere else. (ICAS 1997) 5.0 Students are familiar with conic sections, both analytically and geometrically: 5.1 Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).5.2 Students can take a geometric description of a conic section—for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6—and derive a quadratic equation representing it. 6.0 Students find the roots and poles of a rational function and can graph the function and locate its asymptotes. 7.0 Students demonstrate an understanding of functions and equations defined parametrically and can graph them. Sketch a graph of f(x) = (x - 2)2 -1. Sketch the graphs of g(x) = f(|x|) and of h(x) = |f(x)|. Looking at your graph of h(x), identify a value of x for which h(x + 1) = h(x) -3. 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.
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