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I. Introduction (from Geometry) II. Word Problems and alt trig functions

MENU. TRIGONOMETRY. I. Introduction (from Geometry) II. Word Problems and alt trig functions III. Reference Angles IV. Area and Heron’s Formula V. The Law of Sines VI. The Law of Sines Part 2: The Ambiguous Case VII. Law of Cosines Law of Sines and Cosines practice

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I. Introduction (from Geometry) II. Word Problems and alt trig functions

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  1. MENU TRIGONOMETRY I. Introduction (from Geometry) II. Word Problems and alt trig functions III. Reference Angles IV. Area and Heron’s Formula V. The Law of Sines VI. The Law of Sines Part 2: The Ambiguous Case VII. Law of Cosines Law of Sines and Cosines practice VIII. Radians IX. Unit Circle OPENERSREVIEWSHOMEWORK

  2. OPENERS OPENERS TRIGONOMETRY A B CDEFG HIJKLMNOPQRST Menu

  3. INTRODUCTION to TRIGONOMETRY TRIGONOMETRY Introduction to Trigonometry presentation (from Geometry) Menu

  4. INTRODUCTION to TRIGONOMETRY TRIGONOMETRY Menu

  5. Basic Trig Word Problems TRIGONOMETRY • As a plane flies over East Leyden (2.5 miles away), The angle of elevation to the plane is 720. How high up is the plane? • Godzilla is 400 ft tall. If the angle of elevation to • the top of his head is 480 from where you stand, how far away is Godzilla? Menu

  6. Basic Trig Word Problems TRIGONOMETRY • Superman is flying at an altitude of 1,000 ft. He looks and sees Dr. Evil at an angle of declension (angle of depression) of 730. How far is Superman from Dr. Evil? 4. If the angle of elevation to the sun is 200, and a man casts a 7ft shadow, How tall is he (in feet and inches?). Menu

  7. Basic Trig Word Problems • Jamie flies a kite at an angle of elevation of 480. If she has let out 60 ft of string, how high is the kite? 6. A statue of Mickey Mouse is at the top of a cliff. Bob, standing 100 feet from the cliff notes the angle of elevation to the top of the cliff is 370, and the angle of elevation to the top of the statue is 430. How tall is the s tatue? TRIGONOMETRY Menu

  8. Basic Trig Word Problems TRIGONOMETRY • A giant bird lands on the top of a telephone pole. You are standing 10 feet from the bottom of the pole, and the angle of elevation to the top of the pole is 670. The angle of elevation to the top of the bird is 710. How tall is the bird? 8. From the top of your house, the angle of declension to the close side of the street is 800. The angle of declension to the far side of the street is 760. If your house is 20 ft high, how wide is the street? Menu

  9. Alternative Trig Functions TRIGONOMETRY Menu

  10. Reference Angles Reference Angles TRIGONOMETRY Reference angles Draw a 10 degree angle y We always start from this spot. The positive x axis. 10 x We always go the same direction. Counterclockwise. Menu

  11. Reference Angles TRIGONOMETRY Reference angles Draw a 20 degree angle y x 20 Menu

  12. Reference Angles TRIGONOMETRY Reference angles Draw a 80 degree angle y x 80 Menu

  13. Reference Angles TRIGONOMETRY Reference angles Draw a 120 degree angle y x 120 Menu

  14. Reference Angles TRIGONOMETRY Reference angles Draw a 315 degree angle y x 315 Menu

  15. Reference Angles TRIGONOMETRY Reference angles Draw a 45 degree angle y x 45 Now we are going to use this drawing to find the values of sin, cos and tan for 45 degrees Menu

  16. Reference Angles 45 TRIGONOMETRY y x 45 1. Connect the end of your arrow to the x-axis so it makes a right angle. 2. Fill in the measures with those of a 45-45-90 triangle 3. Identify the sin cos and tan of the triangle you just created. Menu

  17. Use reference angles to find the sin, cos and tan of 60 degrees. Reference Angles TRIGONOMETRY 30 y x 60 1. Connect the end of your arrow to the x-axis so it makes a right angle. 2. Fill in the measures with those of a 30-60-90 triangle 3. Identify the sin cos and tan of the triangle you just created. Menu

  18. Use reference angles to find the sin, cos and tan of 120 degrees. Reference Angles TRIGONOMETRY 30 120 60 y x 1. Connect the end of your arrow to the x-axis so it makes a right angle. 2. Fill in the measures with those of a 30-60-90 triangle 3. Identify the sin cos and tan of the triangle you just created. Menu

  19. Reference Angles TRIGONOMETRY 45 45 y y 135 x x 45 45 Menu

  20. Inverse trig functions Reference Angles TRIGONOMETRY Technically, if they are both more than 180, they add to 540. Two angles with the same sine add to 180 Two angles with the same cosine add to 360 Since the hypotenuse is the longest side, sine and cosine are always between 1 and -1 Menu

  21. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Find the area of this triangle: This formula works for any right triangle, but… Menu

  22. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Find the area of this triangle: Half of the perimeter HERON’s FORMULA Where “S” is the semiperimeter or half the perimeter and a, b, and c are side lengths Menu

  23. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Find the area of this triangle: THIS WORKS IF WE KNOW ALL 3 SIDES, BUT… HERON’s FORMULA Where “S” is the semiperimeter or half the perimeter and a, b, and c are side lengths Menu

  24. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Find the area of this triangle: But you don’t know h. You can figure it out:From the 68, h is the opposite and 8 is the hypotenuse. So use sine! Put these together and get… Menu

  25. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Rather than repeat all this for each problem, we can create a formula It is always 2 sides and the sine of the angle between them. Menu

  26. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Two sides and the angle in between A c b B C a Menu

  27. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY Rather than repeat all this for each problem, we can create a formula 80 20 12 Menu

  28. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY 35 30 70 WE CANT DO THIS ONE! We don’t have 2 sides and the angle between them Menu

  29. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY 35 70 30 Menu

  30. Area of a Triangle and Heron’s Formula AREA TRIGONOMETRY 15 When you’re given the area and two sides, and asked to find the angle between… ? 20 The area of a triangle is 60. Two of the legs are 15 and 20. what is the angle in between them? 60 = ½ 15 x 20 sin (x) 0.4 = sin (x) 15 ? X = sin-1 (0.4) 20 X = 23.6 OR X = 156.4 Menu

  31. Area of a Triangle and Heron’s Formula AREA You can get 2 solutions anytime you are given the area and two sides. TRIGONOMETRY 15 ? 20 Sometimes you can eliminate one of the solutions if you are given another angle, and two angles in the triangle add to more than 180. 15 ? 20 Menu

  32. LAW OF SINES A TRIGONOMETRY b c B C a Menu

  33. LAW OF SINES TRIGONOMETRY The LAW of SINES Menu

  34. LAW OF SINES TRIGONOMETRY Use the law of sines to find x. 23 X 50 72 The LAW of SINES Menu

  35. LAW OF SINES TRIGONOMETRY Find x. 44 30 81 X Menu

  36. LAW OF SINES TRIGONOMETRY Menu

  37. LAW OF SINES Solve this triangle. TRIGONOMETRY A 41 28 62 B C Could we have Used instead? Menu

  38. Solve this triangle. Law of Sines A TRIGONOMETRY 40 16 B C 12 2 Solutions Menu

  39. How can there be 2 solutions? Law of Sines TRIGONOMETRY 2 Solutions 8 10 30 A Menu ANIMATION

  40. Law of Sines TRIGONOMETRY 2 Solutions 8 8 10 10 30 30 A A BOTH of these triangles are solutions Menu ANIMATION

  41. How do I know when there are 2 solutions? Law of Sines TRIGONOMETRY You have to check any time you take the sin -1 2 Solutions 8 8 8 10 10 10 30 30 30 A A A Menu ANIMATION

  42. Solve this triangle. Law of Sines A TRIGONOMETRY 17 55 B C 20 2 Solutions Menu

  43. Solve this triangle. Law of Sines A TRIGONOMETRY 40 52 You don’t have to worry about the second solution here because the angles would add up to more than 180! B C 1 Solution 31 Why does this one have only 1 solution when there are 2 angles with the same sine? Click here or here Menu

  44. Solve this triangle. Law of Sines TRIGONOMETRY B 18 122 12 C A 1 Solution Menu

  45. Solve this triangle. A Law of Sines TRIGONOMETRY 51 24 38 C B No Solution Sine (and Cosine) are NEVER more than 1! What does a triangle with no solution look like? Click here to find out. Menu

  46. LAW of SINES: Summary Law of Sines TRIGONOMETRY When do we use the law of sines? Anytime we are solving a triangle and know an angle and the side across from it. We can use it in a right triangle but usually use sohcahtoa instead Summary When might I get no solutions or 2 solutions Anytime you use sin -1 When using sin -1 we get … 0 solutions 1 solution 2 solutions sin -1 (1.2) The angles in the second set of solutions add to more than 180. You get the second set of solutions when you find the first unknown angle. The missing side is too short to reach the rest of the triangle Menu

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