Mathematical Investigations IV
Trigonometry – Beyond the Right Triangles
Teacher’s Guide
OVERVIEW: Students build immediately upon their knowledge of right triangle Trigonometry that they have encountered in MI II, MI III, or other previous mathematics courses to develop an area formula for all triangles, given side-angle-side (SAS). This is used as a springboard for developing the Law of Sines and exploring the ambiguous case (SSA) via construction. The Law of Cosines follows, as well as sum and differences formulas for sine, cosine and tangent, along with double‑angle formulas. The inverse Trigonometric functions are reintroduced (students saw them in MI-3), which allows them to progress to solving increasingly complex Trigonometric equations and proving Trigonometric identities.
Although several practice problems are provided for each of the formulas in the unit, students will first derive and/or prove them algebraically with guiding questions, building upon their previous experiences with Trigonometry.
Opportunities for extension include various area formulas for triangles, such as Hero's Formula, and half-angle formulas.
PREREQUISITE
KNOWLEDGE: Students are expected to have had experience with basic right triangle Trigonometry, the Trigonometric inverse functions and solving simple Trigonometric equations. A brief review of the domain and range restrictions of the inverse functions is included in this unit, and is critical to students' success in solving more complex equations. Students should also be familiar with the development and use of the Pythagorean Identity, , and the Trigonometric reciprocal identities.
Basic knowledge in compass and protractor use is also assumed in Trig 3.1-4 for the exploration of the ambiguous case.
Perhaps the biggest assumption of prerequisite knowledge for this unit is a certain level of sophistication in algebraic knowledge and skills. Algebraic manipulation (such as factoring and combining or reducing of algebraic fractions) is essential in successfully simplifying and/or proving Trigonometric identities.
CONTENT
OBJECTIVES: Students will
• construct and analyze triangles with partially given information and identify conditions under which a unique triangle exists.
• apply knowledge about right triangle Trigonometry and area of triangles to determine a more general means of finding area of triangles as well as to develop the Law of Sines
• analyze given information about a triangle to determine the appropriate means of solving (Law of Sines, Law of Cosines), including real-world applications
• apply knowledge about inverse Trigonometric functions to solve Trigonometric equations
• apply knowledge and skills of algebra to develop and use Sum and Difference formulas for sine, cosine, and tangent in solving equations and proving Trigonometric identities
• use graphing calculator technology appropriately, know its limitations, and assess the reasonableness of answers given by technology.
COGNITIVE
Objectives: Students will
• investigate and gain insight into basic Trigonometric functions and their graphs by solving complex equations and analyzing the results
• reason inductively and enhance such reasoning through the use of intuition and the building of a foundation of prior experience with basic Trigonometric identities
• communicate effectively using the power and precision of mathematical language and proof.
EXTENSIONS:
• Inclusion of multiple area formulas for triangles, as well as work with radii of inscribed or circumscribed circles about a triangle.
• Inclusion of half-angle formulas for sine, cosine, and tangent.
• Inclusion of extended real-world applications and/or writing assignments.
Lesson 1: Time requirements: 6-7 class periods, 45 – 60 minutes each, although some students may need additional practice time on these concepts throughout the unit.
Materials:
• Trig. 1.1-2, copied back-to-back, one copy per student.
• Trig. 2.1-4, copied back-to-back, one copy per student.
• Trig. 3.1-4, copied back-to-back, one copy per student.
Ø compasses, protractors, rulers
• Trig. 4.1-4, copied back-to-back, one copy per student.
• Trig. 5.1-2, copied back-to-back, one copy per student.
• Trig. 5x.1-2, copied back-to-back, one copy per student (optional).
The purpose of Trig. 1.1-2 is for students to recall their knowledge of right triangles to develop, first numerically, then in general, a formula for area of triangles that applies to all triangles, given SAS. This is the first (of many in this unit) opportunity to emphasize to students the importance of not rounding in an intermediate step of a problem, but rather, to carry exact values until the problem is complete, then round as asked, either by the problem or the teacher's preference.
Trig. 2.1-4 takes this area formula in its three forms (permuting the possibilities of the two given sides and included angle) and guides students through a simple algebraic derivation of the Law of Sines. The remainder of Trig. 2.1-4 provides practice, with the last problem planting the seed of caution necessary for dealing with the SSA case.
Trig. 3.1-4 explores the ambiguous case (SSA), with students constructing triangles with given conditions. Note that the directions for the constructions are on page Trig. 3.3, simply to allow for ease in reading while constructing on page Trig. 3.1. Students will complete the construction and all questions on Trig. 3.3-4 before returning to Trig. 3.2 for problems #3-4. In the construction, it is important for the teacher to watch that students are properly using their protractors and following the directions accurately. The investigative construction exercise, along with questions #1a-k should leave students with sound reasoning behind the lower and upper bounds for the missing side. The geometric underpinnings of this situation, along with probing students to reflect upon the behavior of the sine function over [0, 180°] will give students multiple perspectives behind why SSA is such a special case, as well as how to solve for the second solution. Finally, Problem #4 is challenging, and also introduces NSEW directional notation that will be seen throughout the unit. Working through this problem is usually an ideal time for a large group discussion, concluding the lesson on the Law of Sines.
Trig. 4.1-4 again builds on students' knowledge of right triangle trigonometry. However, since students are now immersed in the Law of Sines, they tend to have difficulty shifting back to this line of thinking without encouragement in order to begin the "Law of Cosines" diagram on Trig. 4.1. Problems #2-7 provide a mixture of practice using the Law of Cosines, although problem #7 can be solved simply with right triangle trigonometry and a system of equations. Students should be encouraged to remember to choose the most appropriate method of solving, and not assume that each problem within a sheet requires the method introduced there. For Problem #8, a reminder about the relationship between an angle bisector and how it divides the intercepted side greatly adds to the efficiency of the solution. Note that it is not necessary to solve for .
This lesson concludes with a collection of practice problems in Trig. 5.1-2. Again, students should be cautioned about rounding within a problem, and/or computing with rounded values. The area of a triangle concept can be extended with Trig. 5x.1-2. Many of the ideas shown here are excellent sources for future problem set questions.
End Notes
This lesson includes review of:
• right triangle trigonometry
• computing the area of a triangle via
• use of compass and protractor
• proving triangle congruence
• the behavior of the sine function over [0, 180°]
• Pythagorean Theorem
• angle bisectors in a triangle
• angle of depression
• volume of a cone
• area of a trapezoid
• domain and range of a function.
This lesson can be extended by inclusions of:
• computer-generated illustrations/animations of the ambiguous case (e.g. Sketchpad )
• introduction of semi-perimeter, inscribed and circumscribed radii, Hero's Formula.
Lesson 2: Time requirements: 6-7 class periods, 45 – 60 minutes each, although some students may need additional practice time on these concepts throughout the unit.
Materials:
• Trig. 6.1-3, copied back-to-back, one copy per student.
• Trig. 7.1-5, copied back-to-back, one copy per student.
• Trig. 8.1-4, copied back-to-back, one copy per student.
• Trig. 9.1-5, copied back-to-back, one copy per student .
• Trig. 10.1-2, copied back-to-back, one copy per student .
• Trig. 10x.1-3, copied back-to-back, one copy per student (optional).
The inverse functions for sine, cosine, and tangent are reviewed quickly in Trig. 6.1-2, including their domains, ranges, and graphs. The general concept of inverse function is also reviewed here, along with trigonometric values of special angles. The subtle, but critical difference between problems #7f and #8 should not be overlooked. This is an excellent occasion to enforce the relevance of the restricted ranges of the inverse trigonometric functions. The last problem on Trig. 6.3 provides an opportunity to refresh students' memories of solving trigonometric equations in the Mathematical Investigations' earlier unit on Trigonometry (or other previous class) over a particular range of values. Incorporating graphing technology to illustrate the window in question and the number of solutions within it can greatly enhance student understanding of the task. This is especially important for a problem of this type that can be solved mechanically, with little or no understanding of the meaning of the solution(s). This concept will be more fully explored in Trig. 12.1-3.
Trig. 7.1-5 also builds on prior experience in trigonometry, developing the three Pythagorean Identities, and the sum/difference formulas for cosines. Note that the last collection of problems on Trig. 7.5 is students' first opportunity in this unit to determine values of trigonometric ratios via reference triangles and the Pythagorean Theorem. A hint in this direction is sometimes needed. It is important to point out the range in which the angles lie in order to determine the appropriate sign of the requested values. Although are both in the first quadrant here, future problems will place them elsewhere.
Although the main purpose of Trig. 8.1-4 is for students to develop and use the sum/difference formulas for sine, the introductory discussion on co-functions is noteworthy. Further problems invite students to analyze the expanded version of the sum/difference formulas for both cosine and sine and recognize them in order to simplify.
Next, the sum/difference formulas for tangent are derived and practiced. Also incorporated into Trig. 9.1-5 are the ideas of determining an angle between two lines and angle of inclination of a line.
The double-angle formulas are seen as a natural extension of the sum formulas, and students have already begun to explore this notion in their earlier work. Practice here is brief, but will be accompanied by practice in the problem sets. This work can also be extended to include the half-angle formulas in Trig. 10x.1-3.
End Notes
This lesson includes review of:
• one-to-one functions and inverses
• inverse trigonometric functions
• trigonometric values of special angles
• solving trigonometric equations over a given domain
• determining period of a trigonometric function from its equation
This lesson can be extended by inclusions of:
• half-angle formulas
• sum/difference formulas for secant, cosecant and cotangent
• graphing of transformed inverse trigonometric functions.
Lesson 3: Time requirements: 5-6 class periods, 45 – 60 minutes each, although some students may need additional practice time on these concepts throughout the unit.
Materials:
• Trig. 11.1-3, copied back-to-back, one copy per student.
• Trig. 12.1-3, copied back-to-back, one copy per student.
Trig. 11.1-3 , "Proving Trigonometric Identities," is a challenge for most students. It may be helpful to hand out Trig. 14.1-2 at this time, so that students have a concise list of all relevant formulas at hand, although ultimately, the expectation is for them to work such proofs without it. It is in this work that the need for strong algebraic manipulation skills is first apparent. Seeing the algebra that lies behind the trigonometric expressions is critical for success. In many cases, the work for the problems alternates between an algebraic manipulation and an application of a trigonometric formula or basic identity. In Trig. 11. 2-3, it is particularly important to guide students in how they write their proofs as much as what they write. Manipulating both sides of the equation assumes that what is to be proved in already true, and therefore, is an unacceptable way to proceed. If students' work ends in a reflexive statement, the teacher can be sure that the student has indeed, manipulated both sides of the equation. Instead, students should choose one side, usually the "more complicated" side (if there is one), and transform it into the other side via algebraic manipulation and/or trigonometric identities and formulas. For problems in which "not valid" in an option, students should be reminded that proof by counterexample is a valid approach. If an equality does not hold for even one value, then the statement cannot be considered an identity, and it is therefore, shown to be invalid. Of course, this method works only to disprove!
In Trig. 12.1-3, it is often the rules of algebra that haunt students, even moreso than the trigonometry! For example, taking the square root (or any even root) of both sides of an equation results in two values (), and they must proceed accordingly. Also, the need for factoring is prevalent in these problems. Students should also be careful not to divide by a variable quantity (e.g. "" in Problem #6), since this could inherently cause division by zero; ultimately, this may eliminate valid solutions, as seen in Problem #6. In contrast, dividing both sides of the equation in Problem #4 by "" does not have the same implications, since the original problem is not defined at precisely the values at which . Students should be expected to solve equations of this type without a calculator when answers are forms of special angles. Otherwise, they should be expected to solve the problem by hand to the point where the inverse trigonometric function keys are the only necessary use of the calculator. Problems #15-17 are more calculator-dependent, and should be addressed accordingly. The use of the graphs and points of intersection to solve Problem #16, for example, is a convenient way for the teacher to help students reflect on how a "solution to an equation" is represented graphically.
End Notes
This lesson includes review of:
• algebraic manipulation (factoring, The Zero Principle, common denominators, etc.)
• 45-45-90 and 30-60-90 right triangle relationships and their trigonometric ratios.
• finding a fixed point of a sequence via recursion (calculator)
• finding points of intersection between two graphs via a graphing calculator.
This lesson can be extended by inclusions of:
• proofs/equation solving that include half-angle formulas
• further explorations into applied situations.
Trigonometry/Teacher’s Guide-8 F06