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Course 2, Unit 7 - Trigonometric Methods

Trigonometry or "the measure of triangles" is an important and useful area of mathematics that naturally connects concepts and methods of geometry and algebra. Trigonometric Methods builds on the Course 1 geometry unit, Patterns in Shape, and the Course 2 unit, Coordinate Methods. In Course 1, students explored the rigidity of triangles and minimal conditions that are sufficient to completely determine a triangle's size and shape. Once a triangle is completely determined, a next logical question is "How can the measures of the triangle's unknown, but rigidly determined, sides and angles be calculated?". Trigonometry provides those methods, namely, trigonometric ratios for right triangles and the Law of Sines and Law of Cosines for any triangle. Interestingly, when angles are extended to include directed angles of rotation, powerful connections between trigonometric ratios and circular motion become evident. Trigonometric methods are extraordinarily useful for solving applied problems in land surveying, engineering, and various applied sciences and for designing and analyzing mechanisms whose function is based on either triangles or circles and their properties. Additionally, the trigonometric ratios, when viewed as functions of the measures of angles in standard position, have many mathematically interesting properties.

Key Ideas from Course 2, Unit 7

  • Trigonometric functions in standard position: If P(xy) is a point (not the origin) on the terminal side of an angle in standard position and r = √(x2 + y2), then the ratios y/x, y/r, and x/r do not depend on the choice of P; they depend only on the measure of ∠POQ. That is, these ratios are functions of the measure θ of the angle. These functions are called trigonometric functions and are given special names as indicated below. (These definitions are developed on pp. 459-462.)

  • Right triangle definitions of sine, cosine, and tangent: sin A = (length of leg opposite angle A)/(length of hypotenuse); cos A = (length of leg adjacent to angle A)/(length of hypotenuse); tan A = (length of leg opposite angle A)/(length of leg adjacent to angle A).

    If you know the lengths of two sides, or the measure of an angle and length of a side, of a right triangle, you can find the lengths of all other sides. For example, if sin A = a/c, then if we know lengths a and c, we can find sin A and then take an inverse sine to find the measure of angle A. If sin A = a/c and we know the measure of angle A and length c, then a = c sin A, so we can find length a. An example using the tangent ratio follows.

    If we know that angle A is 43°, then we find the missing side length a by setting up the ratio tan 43° = opp/adj = a/11. a = 11(tan 43°) ≈ 10.26.

  • Indirect measurement: Trigonometric ratios and values, whether thought of as functions in standard position or as ratios of the lengths of sides of right triangles, can be used to solve problems that require finding lengths that cannot be measured directly. (See the example on p. 469.)

  • Law of Sines:

  • Law of Cosines:

  • SSA triangle conditions: In Course 1 Unit 6, Patterns in Shape, conditions that ensure that a pair of triangles are congruent were explored. The measures of two sides and an angle opposite one of the known sides (also called SSA) do not always determine the size and shape of a unique triangle, but sometimes that information is sufficient. In Lesson 2 Investigation 3, students use the CPMP-Tools custom tool "Explore SSA" and the Law of Sines and the Law of Cosines to explore the conditions under which SSA determines no triangle (if the Triangle Inequality fails), exactly one triangle, and exactly two triangles. (See pp. 498-501 and the custom tool "Explore SSA". Use the Course 2, Geometry menus to find "Explore SSA".)

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