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Chapter 8

Chapter 8. Right Triangles and Trigonometry. Solve for the missing side length in simplest radical form:. Pythagorean Theorem:. Pythagorean Theorem Converse :. Pythagorean Triples: 3 positive integers a, b, c such that . 5, 12, 13. 3, 4, 5. 8, 15, 17. 7, 24, 25.

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Chapter 8

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  1. Chapter 8 Right Triangles and Trigonometry

  2. Solve for the missing side length in simplest radical form:

  3. Pythagorean Theorem: Pythagorean Theorem Converse:

  4. Pythagorean Triples: 3 positive integers a, b, c such that 5, 12, 13 3, 4, 5 8, 15, 17 7, 24, 25 12, 35, 37 11, 60, 61 Any multiple of each triple with also satisfy Example: 6, 8, 10 = 2 X (3, 4, 5)

  5. Theorem 8-3: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.

  6. Theorem 8-4: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

  7. Examples: Determine if the triangle is right, acute, or obtuse. 5, 7, 9 2) 3) 3.1, 4.5, 5.2 4)

  8. Use Pythagorean Theorem to Solve for x:

  9. Special Right Triangles 45°-45°-90° Triangle Theorem: In a 45°-45°-90° Triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg.

  10. Special Right Triangles 30°-60°-90° Triangle Theorem: In a 30°-60°-90° Triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the length of the shorter leg.

  11. Examples:

  12. Find the values of x and y in simplest radical form

  13. Determine what type of triangle each is based on the given side lengths: 1) 5, 9, 4 2) 10, 9, 5 Find the length of the missing side:

  14. 8.3/8.4 Trig Ratios

  15. Tangent Ratio opposite adjacent

  16. Tangent Ratio adjacent opposite

  17. Examples: Find the tangent ratio of angle A and the tangent ration of angle C.

  18. Checkpoint Find Tangent Ratio Find tan Sand tan R as fractions in simplified form and as decimals. Round to four decimal places if necessary. 1. tan S = tan R = ≈1.3333 =0.75; ANSWER 12 3 4 5 =2.4 tan S = tan R = ≈0.4167; 12 3 4 5 ANSWER 2.

  19. Checkpoint Find Tangent Ratio Use a calculator to approximate the value to four decimal places. 3. tan 35° 4. tan 85° 11.4301 0.7002 0.1763 ANSWER ANSWER ANSWER 5. tan 10°

  20. Find the value of x to the nearest tenth.

  21. Find the values of x and y to the nearest tenth.

  22. Find the value of x to the nearest tenth. 1) 2) 3)

  23. Sine Ratio adjacent opposite

  24. Examples: Find the sine ratio of angle A and the sine ratio of angle C.

  25. Cosine Ratio adjacent opposite

  26. Examples: Find the cosine ratio of angle A and the cosine ratio of angle C.

  27. Checkpoint Find Sine and Cosine Ratios Find sin A and cos A. 1. ANSWER 3 8 7 15 4 24 sin A = sin A = ; ; cos A = cos A = 25 17 5 17 5 25 2. ANSWER 3. ; sin A = cos A = ANSWER

  28. Checkpoint Use a Calculator for Sine and Cosine Use a calculator to approximate the value to four decimal places. 7. sin 43° 8. cos 43° 9. sin 15° 0.9659 0.6820 0.2588 0.7314 ANSWER ANSWER ANSWER ANSWER 10. cos 15°

  29. Find the value of x using sine.

  30. Find the value of x using cosine.

  31. Day 2

  32. Warm Up: Find the value of x to the nearest whole.

  33. Checkpoint Find the lengths of the legs of the triangle. Round your answers to the nearest tenth. 15. 16. 17. Find Leg Lengths a ≈ 3.9;b ≈ 5.8 a ≈ 10.9;b ≈ 5.1 a ≈ 3.4;b ≈ 3.7 ANSWER ANSWER ANSWER

  34. Checkpoint 39 Find Sine and Cosine Ratios Find sin A and cosA. Write your answers as fractions and as decimals rounded to four decimal places. ; 9 5 sin A = ≈ 0.9756 40 ANSWER 4. 41 8 41 cos A = ≈ 0.2195 5. ; sin A = ≈ 0.7071 ANSWER 2 2 2 cos A = ≈ 0.7071 2 ; sin A = ≈ 0.7806 6. ANSWER 8 cos A = ≈ 0.625

  35. Find the value of x to the nearest tenth.

  36. Find the value of x.

  37. Find the value of x.

  38. Angles of Elevation and Depression

  39. Angles of Elevation and Depression Angles of elevation and depression are ALWAYS POSITIVE!!! Horizontal Line Angle of Depression They come from the line of sight. Angle of Elevation Horizontal Line

  40. #1 Identifying Angles of Elevation and Depression

  41. Word Problem You sight a rock climber on a cliff at a 32o angle of elevation. The horizontal ground distance to the cliff is 1000 ft. Find the line of sight distance to the rock climber. x 1000 ft

  42. Word Problem An airplane pilots sights a life raft at a 26o angle of depression. The airplane’s altitude is 3 miles. What is the airplane’s surface distance d from the raft? 3 mi d

  43. At Hammonasset Beach, Dave is flying a kite. Currently, the kite string is 30 feet. If Dave is holding the reel about 3 feet from the ground, and the angle of elevation from the reel to the bottom of the kite is 48 degrees, how high is the kite?

  44. To approach a runway, a pilot must begin a 3 degree descent starting from an altitude of 2714 ft. How many miles (ground distance) from the runway is the airplane?

  45. Snoopy is out for a stroll and notices a butterfly resting on top of his dog house. Snoopy is about 2.5 ft high and knows that his dog house is about 3.75 ft high. If Snoopy is about 4 ft from his dog house, what is the angle of elevation from Snoopy to the butterfly?

  46. A blimp is providing aerial television views of a football game. The television camera sights the stadium at a 7 degree angle of depression. The blimp’s altitude is 400 m. What is the line-of-sight distance from the TV camera to the stadium? Round to the nearest meter.

  47. Checkpoint Find Side Length Find the value of x. Round your answer to the nearest tenth. 9. 10. 12.6 10.4 34.6 ANSWER ANSWER ANSWER 11.

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