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Section 4.8 Applications with right Triangles

Section 4.8 Applications with right Triangles. Objective: Solve real world problems with right triangles. Solving Right Triangles. a. a. θ. θ. θ. a. a. a. b. b. a. b. Solving Right Triangles.

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Section 4.8 Applications with right Triangles

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  1. Section 4.8Applications with right Triangles Objective: Solve real world problems with right triangles

  2. Solving Right Triangles a a θ θ θ a a a b b a b Solving Right Triangles Solving a right triangle means to find the lengths of the sides and the measures of the angles of a right triangle. Some information is usually given. • an angle and a sidea, • or two sides, aandb.

  3. 5 30○ 5 opp Since = sin 30 = = , it follows that hyp = 10. hyp hyp To get the last side, note that 5 30○ ; = cos 30 = therefore, adj = Solving A Right Triangle Given an Angle and a Side Solving A Right Triangle Given an Angle and a Side Solve the right triangle. The third angle is 60, the complement of 30. Use the values of the trigonometric functions of 30o. 10 60○

  4. x Post B Post A 100 ft. opp 73○ 3.27 = tan 73= = adj Example 1: Application Example 1: A bridge is to be constructed across a small river from post Ato post B. A surveyor walks 100 feet due south of post A. Shesights on both posts from this location and finds that the anglebetween the posts is 73. Find the distance across the river frompost A to post B. Use a calculator to find tan 73o = 3.27. It follows that x = 327. The distance across the river from post A to post B is 327 feet.

  5. Inverse Trigonometric Functions on a Calculator Inverse Trigonometric Functions on a Calculator Labels for sin1, cos1, and tan1are usually written above the sin, cos, and tan keys. Inverse functions are often accessed by using a key that maybe be labeled SHIFT, INV, or 2nd. Check the manual for your calculator. Example: Find the acute angle  for which cos  = 0.25. Calculator keystrokes: (SHIFT)cos1 0.25 = Display: 75.22487

  6. Solving a Right Triangle Given Two Sides Solve the right triangle shown. 5 θ hyp = = 7.8102496 5 opp tan  = = and  = tan-1( ). adj 6 Solving a Right Triangle Given Two Sides Solve for the hypotenuse: hyp2 = 62 + 52 = 61 6 50.2○ Solve for : 39.8○ Calculator Keystrokes: (SHIFT)tan1( 5  6 )Display: 39.805571 Subtract to calculate the third angle: 90 39.805571 = 50.194428.

  7. Angle of Elevation and Angle of Depression line of sight object observer horizontal horizontal observer line of sight object Angle of Elevation and Angle of Depression When an observer is looking upward, the angle formed by a horizontal line and the line of sight is called the: angle of elevation. angle of elevation When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: angle of depression. angle of depression

  8. Example 2: Application d = = 146.47. Example 2: A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16. What is the distance from the ship to the base of the cliff? observer horizontal 16○ angle of depression cliff42 m line of sight 16○ ship d The ship is 146 m from the base of the cliff.

  9. Example 3: Application sin  = = 0.875 Example 3: A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60 angleof inclination. Does the painter’s plan satisfy the safetyrequirements for the use of the ladder? ladder house 16 14 θ Next use the inverse sine function to find .  = sin1(0.875) = 61.044975 The angle formed by the ladder and the ground is about 61. The painter’s plan is unsafe!

  10. Section 4.4Trigonometric Functionsof General Angles Objectives: Evaluate trig functions of any angle Use reference angles to evaluate trig functions.

  11. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Let θ be any angle in standard position, and let (x,y) denote the coordinates of any point, except the origin (0, 0), on the terminal side of θ. If r = denotes the distance from (0, 0) to (x,y) then the six trigonometric functions of θ are defined as the ratios: provided no denominator equals 0. If a denominator equals 0, that trigonometric function of the angle θ is not defined.

  12. TRIGONOMETRIC FUNCTIONS OF QUADRANTAL ANGLES

  13. COTERMINAL ANGLES Two angles in standard position are said to be coterminal if they have the same terminal side NOTE:Coterminal angles are NOT equal, they merely stop at the same place.

  14. COTERMINAL ANGLES AND TRIGONOMETRIC FUNCTIONS Because coterminal angles have the same terminal side, the values of the six trigonometric functions of coterminal angles are equal.

  15. SIGNS OF THE TRIGONOMETRIC FUNCTIONS

  16. REFERENCE ANGLES Let θ denote a non-acute angle, in standard position, that lies in a quadrant. The acute angle formed by the terminal side of θ and either the positive x-axis or the negative x-axis is called the reference angle for θ.

  17. THE REFERENCE ANGLE THEOREM Reference Angle Theorem: If θ is an angle, in standard position, that lies in a quadrant and α is its reference angle, then where the + or − sign depends on the quadrant in which θlies.

  18. Examples see handout

  19. Assignment • Unit circle Quiz next time • Page 338 • 1-21 all, 31 • Page 297 • 2-24 even, 37-44, 45-57 odd, 85-90

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