More Trigonometry for the Solution of Triangles Something to experiment with: Use your calculator to see what happens when you evaluate these. Make sure you are in DEGREE MODE. a. (SIN(26))^2 +(COS(26))^2 b. (SIN(51))^2 +(COS(51))^2 c. (SIN(167))^2 +(COS(167))^2 d. (SIN(6))^2 +(COS(6))^2 e. (SIN(90))^2 +(COS(90))^2 f. Using inductive thinking, generalize to conclude that sin2 x + cos2 x = ______. Law of Cosines Is this triangle “completely determined”? That means, is there a unique value for the length of the unknown sides and for the measured angles? From the congruent work that we have recently done, the answer is YES! How can we use trigonometry to find the length of the missing side? 20 35 24 Trigonometry WITHOUT right triangles – the Law of Cosines Trigonometry and Triangles page 2 Proof: For what types of problems can the Law of Cosines be used? Exercises with the law of cosines. Find the missing side. 1. 15.78 127.4 2.65 2. Find the measure x and y. y 30 26 Trigonometry and Triangles page 3 x 46 Homework: Law of Cosines Each problem refers to ABC in which side a is opposite A, side b is opposite B, and side c is opposite C. Solve for the indicated part of the triangle. Give answers to two decimal places. 3. a 12, b 14, c 20 Find m C 4. B 72.3, a 78, c 16 Find b. Trigonometry and Triangles page 4 5. a 26.12, b 21.34, c 19.25 Find m A 6. C 28.43, a 6, b 9 Find c. Trigonometry and Triangles page 5 7. The longer base of an isosceles trapezoid measures 14 ft. The nonparallel sides measure 10 ft., and the base angles measure 80. Find the length of a diagonal. 8. Three circles are arranged as shown in the figure below. Find the length of PQ. 1.1 P 109 1.4cm 1.8cm Q 9. A weather balloon is directly west of two observation stations that are 10 miles apart. The angles of elevation of the balloon from the two stations are 17.6 and 78.2. How high is the balloon? Law of Sines What is the Law of Sines? In any triangle, the sides are proportional to the sines of the opposite angles. Proof: For what types of problems can the Law of Sines be used? Trigonometry and Triangles page 7 Find the missing parts of each triangle. 10. 125 122 18 11. 48 98 73 Homework: Law of Sines Find all the missing parts of each triangle. Round lengths to the nearest tenth, and angles to the nearest tenth of a degree. 14. 152 110 28 Trigonometry and Triangles page 8 12. 250 15 120 13. 52 38 47 15. 45 43 58 Trigonometry and Triangles page 9 16. The Leaning Tower of Pisa makes an angle of 5.33 with the vertical. When the angle of elevation of the sun is 36.33, the shadow of the tower (falling on the horizontal plane) is 227 feet long. Find the distance (x) from the bottom to the top of the tower. x 36.33 shadow 5.33 17. Find the length of x rounded to hundredths. 30 40 5 25 x 60 35 55 Trigonometry and Triangles page 10 18. Find AB. B A 33 42 41 C 37 100 cm D The Ambiguous Case – Geometer’s Sketchpad Investigation The “Ambiguous Case” refers to problems in which the given information about a triangle is ______________________________ . For this investigation we will assume we are given angle A, side a, and side b. GOAL: to construct as many triangles as possible with the given information in order to develop some rules for determining how many triangles can be formed. Use Geometer’s Sketchpad. Trigonometry and Triangles page 11 Angle A a < b a > b a = b Acute Obtuse Use the conjectures you reached above to answer this question. How many triangles can be constructed with the given information? 19. A = 30, a = 6, and b = 7 ________ 20. A = 48, a = 24, and b = 20 ________ 21. A = 84, a = 15, and b = 58 ________ 22. A = 124, a = 40, and b = 40 ________ 23. B = 74, a = 68, and b = 70 ________ Solve for all missing parts (lengths and angle measures) of each triangle. Trigonometry and Triangles page 12 24. 10 6 28 25. A 24 , a = 2.4. and b = 3.1 26. A 138 , a = 42.65. and b = 26.95 Trigonometry and Triangles page 13 Homework: Ambiguous Case Find all the missing parts of each triangle. Round answers to one place after the decimal. If more than one triangle can be formed, solve both triangles. If no triangle can be formed, explain why. 27. A 65.2 , a = 125.9, and b = 130.2 Trigonometry and Triangles page 14 28. A 110 , a = 45, and b = 51 29. A 87 , a = 645, and b = 720 30. A 10.5 , a =100, and b = 148 Trigonometry and Triangles page 15 31. Observers are located on the side of a hill at points P and Q. The hill is inclined 32 with the horizontal. The observer at P determines the angle of elevation to a hot-air balloon to be 62. At the same time, the observer at Q measures the angle of elevation to the balloon to be 71. If P is 60 m down the hill from Q, find the distance from Q to the balloon. 32. Two lighthouses (A and B) on a straight shoreline are 120 miles apart. At the same time, an observer at each lighthouse spots a ship cruising along the coastline. The observer at A measures the angle between the shoreline and the ship to be 42.3. The observer at B measures the angle between the shoreline and the ship to be 68.9. Find the shortest distance from the ship to the shore. ship B A Trigonometry and Triangles page 16 33. A tree is growing vertically on a hillside. The angle of inclination of the hill is 22 with the horizontal. When the angle of elevation of the sun is 52, the tree casts a shadow 215 feet down the hill. Find the height of the tree. 22 .