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Trigonometry 10.1

Trigonometry 10.1. Define trigonometry. Label the sides and angles of a right triangle correctly. Find the ratio of the sides in a right triangle. Use trigonometry to find the measures of unknown sides and angles in right triangles.

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Trigonometry 10.1

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  1. Trigonometry 10.1

  2. Define trigonometry. • Label the sides and angles of a right triangle correctly. • Find the ratio of the sides in a right triangle. • Use trigonometry to find the measures of unknown sides and angles in right triangles. • Use a graphing calculator to find the measures of unknown sides and angles. • Solve a right triangle.

  3. Definition Trigonometry is concerned with the relationship between the anglesand sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement.

  4. Triangle Labeling All angles of a triangle are uppercase letters and the sides opposite them are the corresponding lower case letters.

  5. Calculator for Homework Make sure DEG is shown in the top left corner.

  6. Graphing Calculator Using the calculator to evaluate trig functions To evaluate trig functions of acute anglesother than 30, 45, and 60, you will use the calculator. Your calculator has keys marked Sin, Cos, and Tan. **Make sure the MODE is set to the correct unit of angle measure. (Degree vs. Radian)

  7. Calculator Work Find each value using a calculator. Round to the nearest ten-thousandths degrees. .2924 .6820 g. Sin 17° a. Sin 43° .2588 .1045 h. Cos 75° b. Cos 84° i. Tan 26° c. Tan 15° .4877 .2679 .8290 .5878 j. Sin 56° d. Sin 36° .9272 .6428 k. Cos 22° e. Cos 50° l. Tan 43° f. Tan 38° .9325 .7813

  8. Calculator Work Using Inverse Trigonometric Functions to Find Angles Use a calculator to find an angle A in degrees that satisfies sin A  .9677091705. Solution With the calculator in degree mode, we findthat an angle having a sine value of .9677091705 is 75.4º. Write this as sin-1 .9677091705  75.4º.

  9. Calculator Work Find each value using a calculator. Round to the degree. d. Sin B = .2588 62 15 a. Sin A = .8829 60 e. Cos B =.5592 56 b. Cos A = .5 64 f. Tan B = 2.0503 25 c. Tan A = .4663

  10. TrigDefinitions Opposite ---------------- Hypotenuse S-O-H • Sin (angle) = • Cos (angle) = • Tan (angle) = Adjacent ---------------- Hypotenuse C-A-H Opposite ---------------- Adjacent T-O-A

  11. adjacent opposite In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse A We’ll label them a, b, and c and the angles A,B and C. Trigonometric functions are defined by taking the ratios of sides of a right triangle. c b First let’s look at the three basic functions. hypotenuse SINE COSINE TANGENT C B a They are abbreviated using their first 3 letters

  12. hypotenuse hypotenuse opposite opposite adjacent adjacent

  13. Sin, Cos, or Tan? O A O S H C H T A Answer: Tan You know the adjacent and want the opposite. x 35o 7

  14. Sin, Cos, or Tan? O A O S H C H T A Answer: Sin You know the opposite and want the hypotenuse. x 10 40o

  15. Sin, Cos, or Tan? O A O S H C H T A Answer: Cos You know the adjacent and want the hypotenuse. x 35o 20

  16. Sin, Cos, or Tan? O A O S H C H T A Answer: Sin You know the hypotenuse and want the opposite. 12 x 38o

  17. Sin, Cos, or Tan? O A O S H C H T A Answer: Cos You know the hypotenuse and want the adjacent. 100 21o x

  18. Sin, Cos, or Tan? O A O S H C H T A Answer: Sin You know the opposite and the hypotenuse. You want to find the angle. 18 10 o

  19. Sin, Cos, or Tan? O A O S H C H T A Answer: Tan You know the opposite and want the adjacent. 24 37o x

  20. Sin, Cos, or Tan? O A O S H C H T A Answer: Sin You know the opposite and the hypotenuse. And want to know the angle 15 10 o

  21. Sin, Cos, or Tan? O A O S H C H T A Answer: Tan You know the opposite and want the adjacent. 20 42o x

  22. Sin, Cos, or Tan? O A O S H C H T A Answer: Sin You know the opposite and the hypotenuse. You want to find the angle. 400 200 o

  23. SOH CAH TOA Find the values of sin A, cos A, and tan A; sin B, cos B, and tan B in the right triangle. Solution

  24. SOH CAH TOA 10 8 6

  25. 70 θ 24 SOH CAH TOA Find c. a2 + b2 = c2 242 + 702 = c2 5476 = c2 c = 74 74

  26. SOH CAH TOA Find the values of the trigonometric functions for θ. Find a. a2 + 102 = 262 a2 + 100 = 676576 = c2 c = 24 24

  27. To Solve Any Trig Word Problem • Step 1: Draw a triangle to fit problem • Step 2: Label sides from angle’s view • Step 3: Identify trig function to use • Step 4: Set up equation • Step 5: Solve for variable

  28. Assignment 8.3 Practice 1 – 15

  29. Solve the triangle. x y B 16 ft 55 ° A C Solve means to find all angles and all sides. a. Sin 55 = b. Cos 55 = y  13.11 ft x  9.18 ft c. mB = 35

  30. From a point 80m from the base of a tower, the angle from the ground is 28˚. How tall is the tower? x 28˚ 80 Using the 28˚ angle as a reference, we use opposite and adjacent sides. Use tan tan 28˚ = 80 (tan 28˚) = x x ≈ 42.5 m 80 (.5317) = x

  31. A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75˚, how far is the bottom of the ladder from the base of the building? 20 building ladder 75˚ x Using the 75˚ angle as a reference, we use hypotenuse and adjacent side. Use cos cos 75˚ = 20 (cos 75˚) = x x ≈ 5.2 ft 20 (.2588) = x

  32. Find the missing value. x Find the measure of the missing side or hypotenuse for the triangle. c. b. 37 184.08 13.95 a. f. 42.43 14.14 d. 41.04 e.

  33. Find the missing value Find the measure of the missing side or hypotenuse for the triangle. b. 15.45 ft 6651.87 ft a. x d. 16.48 ft c. 137.97 ft

  34. Find the missing value Find the measure of the missing side or hypotenuse for the triangle. b. 135.32 ft 106.48 ft a. c. d. 8398.54 ft 4.95 ft

  35. Find the missing value Find the measure of the missing side or hypotenuse for the triangle. b. c. a. 72.79 m 74.89 ft 445.38 ft f. e. 3090.96 ft 355.77 m d. 524.46 m

  36. Find the missing value. Angle A 6 ft Find the measure of the missing angle. b. 4.76 60 a. c. 15.95

  37. Things to remember. To solve a triangle find all missing sides an angles. Use inverse trigonometric functions to find a missing angle.

  38. Assignment Geometry: 8.3 Practice 16 – 23 Back 13, 14

  39. Angles of Elevation & Depression 10.2

  40. Definitions Angle of elevation is the angle between the horizontal and the line of sight to an object above the horizontal. Angle of depression is the angle between the horizontal and the line of sight to an object below the horizontal.

  41. Angles of Elevation and Depression Top Horizontal Angle of Depression Line of Sight Angle of Elevation Bottom Horizontal Since the two horizontal lines are parallel, by Alternate Interior Angles the angle of depression must be equal to the angle of elevation.

  42. Classify the angles as an angle of elevation or an angle of depression. 1 1 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. 4 4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

  43. Use the diagram to classify the angles as an angle of elevation or depression. 5 5 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. 6 6 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

  44. Classify the angles as an angle of elevation or depression. 6 9 angle of depression angle of elevation

  45. When the sun is 62˚ above the horizon, a building casts a shadow 18 m long. How tall is the building? x 62˚ 18 shadow Using the 62˚ angle as a reference, we use opposite and adjacent side. Use tan tan 62˚ = 18 (tan 62˚) = x x ≈ 33.9 m 18 (1.8807) = x

  46. A kite is flying at an angle of elevation of about 55˚. Find the height of the kite if 85m of string has been let out. kite 85 x string 55˚ Using the 55˚ angle as a reference, we use hypotenuse and opposite side. Use sin sin 55˚ = 85 (sin 55˚) = x 85 (.8192) = x x ≈ 69.6 m

  47. A 5.50 foot person standing 10 feet from a street light casts a 14 foot shadow. What is the height of the streetlight? 5.5 x˚ 10 14 shadow tan x˚ = About 9.4 ft. x° ≈ 21.45°

  48. The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder? 38º angle of depression 25 Alternate Interior Angles are congruent 38º x Using the 38˚ angle as a reference, we use opposite and adjacent side. Use tan tan 38˚ = 25/x (.7813) = 25/x x = 25/.7813 x ≈ 32.0

  49. Jody sees a plane above the airport at an angle of elevation of 32°. She is 2 miles from the airport where it is circling. How high is the airplane above the airport? The plane is approximately 1.25 miles above the airport. P x 32° J A 2 mi

  50. A forestry service has two fire towers located 8,000 feet apart. If the first is located 1,000 feet above on a mountain, what is the angle of depression from the first to the second tower? T1 X° 1,000 ft. 8,000 ft T2 The angle of depression from the first tower to the second is about 7.18°.

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