1 / 83

Chapter 1

Chapter 1. Trigonometric Functions. 1.1. Angles. 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.

ganit
Download Presentation

Chapter 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 1 Trigonometric Functions

  2. 1.1 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. Naming Angles • Unless it is ambiguous as to the meaning, angles may be named only by a single letter (English or Greek) displayed at vertex or in area of rotation between initial and terminal sides • Angles may also be named by three letters, one representing a point on the initial side, one representing the vertex and one representing a point on the terminal side (vertex letter in the middle, others first or last)

  6. 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

  7. Angle Measures and Types of Angles • The most common unit for measuring angles is the degree. (One rotation = 360o) • ¼ rotation = 90o, ½ rotation = 180o, • Angle and measure of angle not the same, but it is common to say that an angle = its measure • Types of angles named on basis of measure:

  8. Complementary and Supplementary Angles • Two positive angles are called complementary if the sum of their measures is 90o • The angle that is complementary to 43o = • Two positive angles are called supplementary if the sum of their measures is 180o • The angle that is supplementary to 68o =

  9. 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

  10. 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

  11. Portions of Degree: Minutes, Seconds • One minute, 1’, is 1/60 of a degree. • One second, 1”, is 1/60 of a minute.

  12. Perform the calculation. Since 86 = 60 + 26, the sum is written: Perform the calculation. Hint write: Example: Calculations

  13. Convert Convert 34.624 Converting Between Degrees, Minutes and Seconds and Decimal Degrees

  14. 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.

  15. Quadrantal Angles • 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.

  16. Coterminal Angles • A complete rotation of a ray results in an angle measuring 360. Given angle A, and continuing the rotation by a multiple of 360 will result in a different angle, A + n360,with the same terminal side: coterminal angles.

  17. 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)

  18. Homework • 1.1 Page 6 • All: 6 – 9, 14 – 17, 24 – 29, 32 – 35, 38 – 41, 46 – 51, 55 – 58 , 75 – 79 • MyMathLab Assignment 1 for practice • MyMathLab Homework Quiz 1 will be due for a grade on the date of our next class meeting!!!

  19. 1.2 Angle Relationships and Similar Triangles

  20. Q R M N P Vertical Angles • When lines intersect, angles opposite each other are called vertical angles • Vertical angles in this picture: • How do measures of vertical angles compare? Vertical Angles have equal measures.

  21. 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.

  22. 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

  23. Find the measure of each marked angle, given that lines m and n are parallel. What is the relationship between these angles? Alternate exterior with equal measures Measure of each angle? 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

  24. Angle Sum of a Triangle • The instructor will ask specified students to draw three triangles of distinctly different shapes. All the angles will be cut off each triangle and placed side by side with vertices touching. • What do you notice when you sum the three angles? • The sum of the measures of the angles of any triangle is 180.

  25. 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

  26. Types of Triangles: Named Based on Angles

  27. Types of Triangles: Named Based on Sides

  28. Similar and Congruent Triangles • Triangles that have exactly the same shape, but not necessarily the same size are similartriangles • Triangles that have exactly the same shape and the same size are called congruent triangles

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

  30. 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: Measure of D: Angle E corresponds to angle: Measure of E: D A 112 35 E F 112 33 C B Example: Finding Angle Measures on Similar Triangles

  31. 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 on Similar Triangles

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

  33. Homework • 1.2 Page 14 • All: 3 – 7, 9 – 13, 16 – 19, 25 – 36, 41 – 44, 46 – 49, 51 – 54, 57 – 60, 65 – 66, 69 – 70 • MyMathLab Assignment 2 for practice • MyMathLab Homework Quiz 2 will be due for a grade on the date of our next class meeting!!!

  34. 1.3 Trigonometric Functions

  35. Trigonometric Functions Compared with Algebraic Functions • Algebraic functions are sets of ordered pairs of real numbers such that every first member, “x”, is paired with exactly one second member, “y” • Trigonometric functions are sets of ordered pairs such that every first member, an angle, is paired with exactly one second member, a ratio of real numbers • Algebraic functions are given names like f, g or h and in function notation, the second member that is paired with “x” is shown as f(x), g(x) or h(x) • Trigonometric functions are given the names, sine, cosine, tangent, cotangent, secant, or cosecant, and in function notation, the second member that is paired with the angle “A” is shown as sin(A), cos(A), tan(A), cot(A), sec(A), or csc(A) –(sometimes parentheses are omitted)

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

  37. Values of Trig Functions Independent of Point Chosen • For the given angle, if point (x1,y1) is picked and r1 is calculated, trig functions of that angle will be ratios of the sides of the triangle shown in blue. • For the same angle, if point (x2,y2) is picked and r2 is calculated, trig functions of the angle will be ratios of the triangle shown in green • Since the triangles are similar, ratios and trig function values will be exactly the same

  38. (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 .

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

  40. Trigonometric Functions of Coterminal Angles • Note: To calculate trigonometric functions of an angle in standard position it is only necessary to know one point on the terminal side of that angle, and its distance from the origin • In the previous example six trig functions of the given angle were calculated. All angles coterminal with that angle will have identical trig function values • ALL COTERMINAL ANGLES HAVE IDENTICAL TRIGONOMETRIC FUNCTION VALUES!!!!

  41. Equations of Rays with Endpoint at Origin: • Recall from algebra that the equation of a line is: • If a line goes through the origin its equation is: • To get the equation of a ray with endpoint at the origin we write an equation of this form with the restriction that:

  42. 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

  43. From previous calculations: Use the definitions of the trig functions: Example: Finding Function Values continued

  44. Finding Trigonometric Functions of Quadrantal Angles • A point on the terminal side of a quadrantal angle always has either x = 0 or y = 0 (x = 0 when terminal side is on y axis, y = 0 when terminal side is on x axis) • Since any point on the terminal side can be picked, choose x = 0 or y = 0, as appropriate, and choose r = 1 • The remaining x or y will then be 1 or -1

  45. Example: Function Values Quadrantal Angles • Find the values of the six trigonometric functions for an angle of 270. • Which point should be used on the terminal side of a 270 angle? • We choose (0, 1). Here x = 0, y = 1 and r = 1. • Value of the six trig functions for this angle:

  46. Undefined Function Values • If the terminal side of a quadrantal angle lies along the y-axis, then, because x = 0, the tangent and secant functions are undefined: • If it lies along the x-axis, then, because y = 0, the cotangent and cosecant functions are undefined.

  47. 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

  48. Finding Trigonometric Functions of Specific Angles • Until discussing trigonometric functions of specific quadrantal angles such as 90o, 180o, etc., we have found trigonometric functions of angles by knowing or finding some point on the terminal side of the angle without knowing the measure of the angle • At the present time, we know how to find exact trigonometric values of specific angles only if they are quadrantal angles • In the next chapter we will learn to find exact trigonometric values of 30o, 45o, and 60o angles • In the meantime, we can find approximate trigonometric values of specific angles by using a scientific calculator set in degree mode

  49. Finding Approximate Trigonometric Function Values of Sine, Cosine and Tangent • Make sure your calculator is set in degree mode • Depending on your calculator, • Enter the angle measure first then press the appropriate sin, cos or tan key to get the value • Press the sin, cos, or tan key first, then enter the angle measure • Practice on these:

  50. Exponential Notation and Trigonometric Functions • A trigonometric function defines a real number ratio for a specific angle, for example “sin A” is the real number ratio assigned by the sine function to the angle “A” • Since “sin A” is a real number it can be raised to any rational number power, such as “2” in which case we would have “(sin A)2” • However, this value is more commonly written as “sin2 A” sin2 A = (sin A)2 • Using this reasoning then if “tan A = 3”, then: tan4 A =

More Related