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Chapter 7

Chapter 7. Applications of Trigonometry. Section 7.1. Inverse Trigonometric Functions. Definition. The inverse sine function , denoted by sin -1 , is defined as follows: y = sin -1 x  x = sin y and. Definition. The inverse cosine function , denoted by cos -1 , is defined as follows:

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Chapter 7

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  1. Chapter 7 Applications of Trigonometry

  2. Section 7.1 Inverse Trigonometric Functions

  3. Definition • The inverse sine function, denoted by sin-1, is defined as follows: y = sin-1x  x = sin y and .

  4. Definition • The inverse cosine function, denoted by cos-1, is defined as follows: y = cos-1x  x = cos y and .

  5. Remarks • The domain for both sin-1x and cos-1x is the set [-1,1]. • The range for sin-1x is [-/2,/2] while the range for cos-1x is [0,].

  6. Examples • Find the exact values of the following without using a calculator: (1) (2) (3) (4) (5)

  7. Definition • The inverse tangent function, denoted by tan-1, is defined as follows: y = tan-1x  x = tan y and .

  8. Remark • The domain of tan-1x is the set of real numbers and its range is (-/2,/2). • The graph of tan-1x has two horizontal asymptotes, namely y = -/2 and y = /2. • For any real number x, we define cot-1x = /2 – tan-1x.

  9. Definition • The inverse secant function, denoted by sec-1, is defined as follows: y = sec-1x  x = sec y and .

  10. Remark • The domain of sec-1x is [1,+)  (-,-1] and its range is [0,/2)  [,3/2). • For x  (-,-1]  [1,+), we define csc-1x as /2 – sec-1x.

  11. Example 1 • Find the exact values of the following without using a calculator: (a) (b) (c)

  12. Example 2 • Suppose 0 < , < /2. If and , prove that  +  = /4.

  13. Section 7.2 Trigonometric Equations

  14. Introduction • Recall that a trigonometric equation is an identity if the equation is always true whenever the expressions on both sides are defined. • Not all trigonometric equations are identities. • cos 2x = 1/2 • 2cos2x = sin x + 1 • In this section, we shall discuss solutions to such trigonometric equations.

  15. Example 1 • Find the solution set in R of the following equations: (a) (b) 2cos t + csc t = 0 (c) 9cos2 = 1 + 6sin

  16. Example 2 • Find the solutions of the following equations in the interval [0,2): (a) (b)

  17. Example 3 • Find the corresponding degree measures of all solutions of the equation tan2(3x) + 3sec(3x) + 3 = 0

  18. Exercises / Assignment Items • Find the solution set (in radians) of the ff. equations in the indicated intervals: (1) cot2 + 3csc = -3 (2) cos(4x) = 10cos2(2x) – 3, (3) 4csc2x – 3 = 0, [0,2) (4) (5) 2cos(2x)sin(2x) – sin(2x) = 0 (in degrees)

  19. Section 7.3 Solving Right Triangles

  20. Definition • To solve a triangle is to find the unknown measures of its sides and its angles. • In this section, we shall focus only on solving right triangles.

  21. Recall: Trigonometric Functions • If  is an acute angle in a right triangle, then • We can find cot , sec  and csc  by taking the reciprocals of the above.

  22. Theorem • Consider the right triangle where the lengths of the sides and angle are indicated below: Then sin  = a/c, sin  = b/c cos  = b/c, cos  = a/c tan  = a/b, tan  = b/a.

  23. Examples • Solve the ff. right triangles where  = 90o: (a)  = 19.35o, c = 1.86 (b) a = 27.3, c = 35.4

  24. Definition • The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer’s eye (the line of sight).

  25. Definition • If the object is below the level of the observer, then the angle between the horizontal and the observer’s line of sight is called the angle of depression.

  26. Example 1 • At 9:00 AM, a building 200 meters high casts a shadow 38 meters long. What is the angle of elevation of the sun?

  27. Example 2 • Observer A records an angle of depression of 35o50’ from his position on the tenth floor of a building to a car down below. Observer B is located on the seventh floor and records an angle of depression of 23o40’ to the same car. If the distance from Observer A to Observer B is 24 feet, how far is the car from the base of the building?

  28. Definition • In navigation, the course of a ship or airplane is the angle measured in degrees clockwise from the north to the direction in which the carrier is traveling. • The bearing of a particular location P from an observer at O is the angle measured in degrees clockwise from the north of the line segment OP.

  29. Example • A ship leaves port and sails for 4 hours on a course of 78o at 18 knots. Then, the ship changes its course to 168o and sails for 6 hours at 16 knots. At a time of 10 hours after leaving port, find (a) the distance of the ship from the port; (b) the bearing from the port to the ship; (c) the bearing from the ship to the port.

  30. Section 7.4 Law of Sines

  31. Theorem (Sine Law)* • Let ABC have interior angles of measures , , and  at the vertices A, B, and C, respectively. Let the lengths of the sides opposite , , and  be a, b, and c, respectively. Then

  32. Remarks • The sine law is used to solve triangles where the given information is either • Two interior angles and the length of one side; or • The lengths of two sides and the measure of an angle opposite to one of these sides. • For the second case, there may be one, two, or no triangles satisfying the given conditions. This is called the ambiguous case.

  33. Remark • Given two interior angles and the length of a side of a triangle, we can solve the triangle easily using the Sine Law.

  34. Example • Solve the ff. triangle:  = 100.49o,  = 21.72o, a = 725

  35. Remark • Given the lengths of two sides and the measure of an angle opposite one of those sides, there may be one, two, or no triangles satisfying the given measurements. • This is called the ambiguous case.

  36. Solving the Ambiguous Case (without memorizing the cases!) • Suppose that a, b, and  are given. • Step 1: Use the Law of Sines to find . If the value of sin  > 1, then there are no possible triangles. • Step 2: Determine the value of  in Quadrant II. That is,  = 180 – Quadrant I value. • Step 3: Find the missing angle  for both values of  found in Step 2 (using the fact that the sum of the angles of a triangle is 180o). • Step 4: If it is possible to find a valid measure of  for both values of , then two triangles are possible. Otherwise, only one triangle is possible.

  37. Example 1 • Solve the following triangles: (a) c = 58.15, a = 45,  = 67.3o (b) b = 8, a = 9,  = 57.18o (c) c = 4500, b = 4738,  = 75.22o

  38. Example 2 • A pole leans away from the sun at an angle of 7o with the vertical. When the angle of elevation of the sun is 51o, the pole casts a shadow 47 feet long on level ground. How long is the pole?

  39. Section 7.5 Law of Cosines

  40. Theorem (Cosine Law)* • Let ABC have sides a, b, and c with interior angles , ,  opposite a, b, and c, respectively. Then a2 = b2 + c2 – 2bc cos b2 = a2 + c2 – 2ac cos c2 = a2 + b2 – 2ab cos

  41. Remark • The cosine law is used to solve triangles where the given information is either • The lengths of two sides and the measure of their included angle. • The lengths of the three sides. • If the measures of the three sides are given, use the Cosine Law to find the biggest angle first. We can then use the Sine Law to find the two remaining angles.

  42. Example 1 • Solve the following triangles: (a) a = 136, b = 215,  = 117o45’ (b) a = 2.87, b = 3.21, c = 2.13

  43. Example 2 • A man placed a 7.15 foot pole vertically on a hillside with a uniform slope. Then, he measured 101.8 feet from the base of the pole to a point further up the hillside, and 99.2 feet to the same point from the top of the pole. What was the angle of elevation of the hillside?

  44. Example 3 • The adjacent sides of a parallelogram measure 6 feet and 10 feet, and make an angle of 120o with each other. Find the length of the larger diagonal. Then, find the area of the parallelogram.

  45. Summary: Solving Triangle • If the triangle is a right triangle, use the trigonometric ratios. • If not, we have the ff. cases: • Given 2 angles, 1 side: Use the Sine Law. • Given 2 sides and an angle opposite one of the sides: Use the Sine Law. • Given 2 sides and the included angle: Use the Cosine Law. • Given all 3 sides: Use the Cosine Law

  46. Exercises / Assignment Items • Page 338, #s 6, 7, 8

  47. Section 7.6 Polar Coordinates

  48. Motivation • A point P in the rectangular coordinate system is completely specified by its coordinates x and y. • However, there are other coordinate systems that can be used to specify the position of a point P on the plane. One of these is the polar coordinate system.

  49. Polar Coordinates /2-axis y P(x,y) P(r,) r Polar Axis y Pole  x x

  50. Remarks • The angle  is a directed angle; that is, it is positive if it is measured counterclockwise from the initial side to the terminal side, and negative if it is measured clockwise. • The value r is a directed distance, it is positive if the point P lies on the terminal side of  and negative if P is on the extension of the terminal side.

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