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Chapter 1

Chapter 1. Trigonometric Functions. 1.1 Angles. Objective: Understand and apply the basic terminology of angles Warm up : Define and draw a picture of each of the following terms Line Line Segment Ray Right angle Acute angle Obtuse angle

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Chapter 1

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  1. Chapter 1 Trigonometric Functions

  2. 1.1 Angles Objective: Understand and apply the basic terminology of angles Warm up : Define and draw a picture of each of the following terms Line Line Segment Ray Right angle Acute angle Obtuse angle Complementary Angles Supplementary Angles

  3. A B A B A B Basic Terms • Two distinct points determine a line called line AB. • Line segment AB—a portion of the line between A and B, including points A and B. • Ray AB—portion of line AB that starts at A and continues through B, and on past B.

  4. Angle-formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle. Basic Terms continued

  5. Positive angle: The rotation of the terminal side of an angle counterclockwise. Negative angle: The rotation of the terminal side is clockwise. Basic Terms continued

  6. Types of Angles • The most common unit for measuring angles is the degree.

  7. k+20 k 16 Example: Complementary Angles • Find the measure of each angle. • Since the two angles form a right angle, they are complementary angles. Thus, The two angles have measures of 43 + 20 = 63 and 43  16 = 27

  8. 6x + 7 3x + 2 Example: Supplementary Angles • Find the measure of each angle. • Since the two angles form a straightangle, they are supplementary angles. Thus, These angle measures are 6(19) + 7 = 121 and 3(19) + 2 = 59

  9. Degree, Minutes, Seconds • One minute is 1/60 of a degree. • One second is 1/60 of a minute.

  10. Perform the calculation. Since 86 = 60 + 26, the sum is written Perform the calculation. Write Example: Calculations

  11. Convert to decimal degrees. Convert to degrees, minutes, and seconds 36.624 Example: Conversions

  12. Standard Position • An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. • Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles.

  13. Coterminal Angles • A complete rotation of a ray results in an angle measuring 360. By continuing the rotation, angles of measure larger than 360 can be produced. Such angles are called coterminal angles.

  14. o o - 1115 3(360 ) o = 35 Example: Coterminal Angles • Find the angles of smallest possible positive measure coterminal with each angle. • a) 1115 b) 187 • Add or subtract 360 as may times as needed to obtain an angle with measure greater than 0 but less than 360. • a) b) 187 + 360 = 173

  15. Homework • Page 7 # 14 - 42

  16. 1.2 Objective: Compare Angle Relationships and to identify similar triangles and calculate missing sides and angles.

  17. Warm up: Use the graph at the right to find the following 1. Name a pair of vertical angles. 2. Line a and b are what kind of lines. 3. Name a pair of alternate interior angles. 1 2 a 4. Name a pair of alternate exterior angles 3 4 5. Name a pair of corresponding angles. b 5 6 7 8 6. Find the measure of all the angles.

  18. q m n Name Angles Rule Alternate interior angles 4 and 5 3 and 6 Angles measures are equal. Alternate exterior angles 1 and 8 2 and 7 Angle measures are equal. Interior angles on the same side of the transversal 4 and 6 3 and 5 Angle measures add to 180. Corresponding angles 2 & 6, 1 & 5, 3 & 7, 4 & 8 Angle measures are equal. Angles and Relationships

  19. Q R M N P Vertical Angles • Vertical Angles have equal measures. • The pair of angles NMP and RMQ are vertical angles.

  20. q Transversal m parallel lines n Parallel Lines • Parallel lines are lines that lie in the same plane and do not intersect. • When a line q intersects two parallel lines, q, is called a transversal.

  21. Find the measure of each marked angle, given that lines m and n are parallel. The marked angles are alternate exterior angles, which are equal. One angle has measure 6x + 4 = 6(21) + 4 = 130 and the other has measure 10x 80 = 10(21)  80 = 130 (6x + 4) m n (10x 80) Example: Finding Angle Measures

  22. Angle Sum of a Triangle • The sum of the measures of the angles of any triangle is 180.

  23. The measures of two of the angles of a triangle are 52 and 65. Find the measure of the third angle, x. Solution The third angle of the triangle measures 63. 65 x 52 Example: Applying the Angle Sum

  24. Types of Triangles: Angles

  25. Types of Triangles: Sides

  26. Conditions for Similar Triangles • Corresponding angles must have the same measure. • Corresponding sides must be proportional. (That is, their ratios must be equal.)

  27. Triangles ABC and DEF are similar. Find the measures of angles D and E. Since the triangles are similar, corresponding angles have the same measure. Angle D corresponds to angle A which = 35 Angle E corresponds to angle B which = 33 D A 112 35 E F 112 33 C B Example: Finding Angle Measures

  28. Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF. To find side DE. To find side FE. D 16 A 112 35 64 E F 32 112 33 C B 48 Example: Finding Side Lengths

  29. A lighthouse casts a shadow 64 m long. At the same time, the shadow cast by a mailbox 3 feet high is 4 m long. Find the height of the lighthouse. The two triangles are similar, so corresponding sides are in proportion. The lighthouse is 48 m high. 3 4 x 64 Example: Application

  30. Homework • Page 14-16 # 3-13 odd, 25-35 odd, 45-56 odd

  31. 1.3 Objective: To understand and apply the 6 trigonometric functions

  32. Warm up • In the figure below, two similar triangles are present. Find the value of each variable. x-2y 5 74 x-5 x+y 10 74 15

  33. Trigonometric Functions • Let (x, y) be a point other the origin on the terminal side of an angle  in standard position. The distance from the point to the origin is The six trigonometric functions of  are defined as follows.

  34. (12, 16) 16  12 Example: Finding Function Values • The terminal side of angle  in standard position passes through the point (12, 16). Find the values of the six trigonometric functions of angle .

  35. Example: Finding Function Values continued • x = 12y = 16r = 20

  36. Find the six trigonometric function values of the angle  in standard position, if the terminal side of  is defined byx + 2y = 0, x  0. We can use any point on the terminal side of  to find the trigonometric function values. Example: Finding Function Values

  37. Choose x = 2 The point (2, 1) lies on the terminal side, and the corresponding value of r is Use the definitions: Example: Finding Function Values continued

  38. Example: Function Values Quadrantal Angles • Find the values of the six trigonometric functions for an angle of 270. • First, we select any point on the terminal side of a 270 angle. We choose (0, 1). Here x = 0, y = 1 and r = 1.

  39. Undefined Function Values • If the terminal side of a quadrantal angle lies along the y-axis, then the tangent and secant functions are undefined. • If it lies along the x-axis, then the cotangent and cosecant functions are undefined.

  40. sin  cos  tan  cot  sec  csc  0 0 1 0 undefined 1 undefined 90 1 0 undefined 0 undefined 1 180 0 1 0 undefined 1 undefined 270 1 0 undefined 0 undefined 1 360 0 1 0 undefined 1 undefined Commonly Used Function Values

  41. Homework • Page 25 # 18-46 even

  42. 1.4 Objective: to apply the definitions of the trigonometric functions

  43. Warm-up • What is the reciprocal of • 2/3? • 1 2/5? • 0? • Cos 0? • Sin 0? • Tan 0?

  44. Reciprocal Identities

  45. cos  if sec  = Since cos  is the reciprocal of sec  sin  if csc  Example: Find each function value.

  46.  in Quadrant sin  cos  tan  cot  sec  csc  I + + + + + + II +     + III   + +   IV  +   +  Signs of Function Values

  47. Example: Identify Quadrant • Identify the quadrant (or quadrants) of any angle  that satisfies tan  > 0, cot  > 0. • tan  > 0 in quadrants I and III • cot  > 0 in quadrants I and III • so, the answer is quadrants I and III

  48. Ranges of Trigonometric Functions • For any angle  for which the indicated functions exist: • 1. 1  sin   1 and 1  cos   1; • 2. tan  and cot  can equal any real number; • 3. sec   1 or sec   1 and csc   1 or csc   1. (Notice that sec  and csc  are never between 1 and 1.)

  49. Pythagorean Quotient Identities

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