TRIGONOMETRY. circular functions. These functions, especially the sine and

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1 TRIGONOMETRY Trigonometry has its origins in the study of triangle measurement. Natural generalizations of the ratios of righttriangle 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. Students studying trigonometry should explore data from such real world phenomena, and also should identify and analyze the corresponding trigonometric models. The study of inverse trigonometry functions; trigonometry equations and identities; the Law of Sines and the Law of Cosines; vectors; and polar coordinates should also be included in the course. It is important to note that all of the objectives and competencies listed in this outline have also been included in the list of competencies for Pre-Calculus. Please refer to the general introduction to the Indiana High School Mathematics Competencies (pp. i - iv) for more information on the structure of the competency lists, modes of instruction and the use of technology. Scientific calculators and graphing technology play an important role in the development of students' understanding of the concepts of trigonometry as well as providing students with more time and power to explore realistic applications. Trigonometry - 162

2 OBJECTIVE 1 The learner will demonstrate proficiency in using scientific calculators and graphing technology. Competency 1.1 Graph functions and relations. 1.2 Analyze functions. (Zeros, domain, range, asymptotes, etc.) 1.3 Solve equations. 1.4 Evaluate trigonometric and inverse trigonometric expressions. 1.5 Verify trigonometric identities. 1.6 Convert radians/degrees. Trigonometry - 163

3 OBJECTIVE 2 The learner will apply trigonometry to problem situations Competency 2.1 Solve real world You are given the job of measuring the height of a mountain. Standing on level problems involving right ground, you measure the angle of elevation to the top to be 17.3 degrees. You now triangle trigonometry. move 350 meters closer and find the angle of elevation to be 29.1 degrees. How high is the mountain? A helicopter is approaching Indianapolis International Airport at an altitude of 3 km. The angle of depression from the helicopter to the airport is degrees. a) What is the helicopter's ground distance to the airport? b) What is the distance the helicopter will actually travel as it descends along the line of sight? 2.2 Define trigonometric functions using right triangles. 2.3 Apply the laws of sines and cosines to the solution of application problems. Trigonometry - 164

4 Competency 2.4 Define vectors and use in Bill's ship is traveling on a bearing of 112 degrees at 29 knots. The current is on a solving real world bearing of 200 degrees at 7 knots. Find the actual bearing and speed of the ship. problems A plane is flying on a heading of 310º at a speed of 320 miles per hour. The wind is blowing directly from the east at a speed of 32 miles per hour. a) Make a drawing to indicate the plane's motion. b) What equations model the plane's motion (without the wind)? c) Make a drawing to indicate the wind's motion. d) What equations model the wind's motion? e) What are the resultant equations modeling the plane with the wind? f) After 5 hours, where is the plane? g) Carefully describe how to find the angle adjustment the pilot must make so that the plane is not blown off course during a 5 hour flight. h) Write the final equations which combine both plane and wind motion contributions to the flight. i) Test your equations by graphing and tracing. Trigonometry - 165

5 OBJECTIVE 3 The learner will develop an understanding of trigonometric functions. Competency 3.1 Solve real world A creature from Venus lands on Earth. Its body temperature varies sinusoidally with problems involving time. Twenty-five minutes after landing it reaches a high of 130 degrees Fahrenheit. applications of Fifteen minutes after that it reaches its next low of 100 degrees Fahrenheit. trigonometric functions. a) Sketch the graph of this function. b) Write an equation expressing temperature in terms of minutes after landing. c) What was the temperature upon landing? d) Find the first two times the temperature was 117 degrees. 3.2 Develop the relationship Hold your arm out at eye level. Have someone measure the distance from your eye to between degree and the end of your fingertips. Cut a strip of paper that length. Now, with your arms radian measures. outstretched, hold the paper parallel to your body, with one hand on each end of the strip. The angle your arms form approximates a radian. Why? Approximately how many degrees is your angle? 3.3 Define trigonometric functions using the unit circle. 3.4 Learn exact sine, cosine, tangent values for multiples of B, B 2, B 3, B 4, B 6, 0. Trigonometry - 166

6 Competency 3.5 Find domain, range, intercepts, periods, amplitudes, and asymptotes of the trigonometric functions. 3.6* Identify odd and even functions and the implications for their graphs. * Optional 3.7 Graph trigonometric Graph y = 4-3 cos (2x - 40). Identify the period, amplitude, phase shift and vertical functions and analyze displacement. graphs of trigonometric functions. 3.8 Define and evaluate inverse trigonometric functions. 3.9 Analyze and graph translations of trigonometric functions. Trigonometry - 167

7 Competency 3.10 Make connections among Given the right triangle below, the right triangle ratios, trigonometric functions, a) find the exact sine, cosine and tangent of angle A. and circular functions. b) find the real numbers x, 0 # x < 2B, with exactly the same sine, cosine, and tangent values. Trigonometry - 168

8 OBJECTIVE 4 The learner will solve trigonometric equations and verify trigonometric identities. Competency 4.1 Solve real world problems A baseball player hits the ball and the ball leaves the bat at an angle of x with a involving applications of velocity, v, of 100 ft/sec. The ball is caught 350 ft away from home plate. Use the trigonometric equations. equation below to find the angle of projectile (x) of the ball. d = 1 32 (V sin2x) A gun with a muzzle velocity of 1200 ft/sec is pointed at a target 2000 yards away. Excluding air resistance, what should be the minimum angle of elevation of the gun? d = 1 32 (V sin2x) 4.2 Apply the fundamental trigonometric identities. 4.3 Use the fundamental identities to verify simple identities. Trigonometry - 169

9 Competency 4.4 Use graphing technology Use a graphing calculator to solve for real numbers, x, 0 # x <2B. Discuss the results for solving trigonometric of your findings. equations. a) cos x = -1 b) cos 2x = -1 c) cos 3x = -1 d) 6 sin 2x = 5 (y = 6 sin 2x and y = 5) Trigonometry - 170

10 OBJECTIVE 5 The learner will understand the connections between trigonometric functions and polar coordinates and complex numbers. Competency 5.1 Define polar coordinates and relate polar to Cartesian coordinates. 5.2 Graph equations in the polar coordinate plane. 5.3 Define complex numbers and convert to trigonometric form. 5.4* State, prove, and use DeMoivre's Theorem. * Optional Trigonometry - 171