MAC Module 1 Trigonometric Functions. Rev.S08

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1 MAC 1114 Module 1 Trigonometric Functions

2 Learning Objectives Upon completing this module, you should be able to: 1. Use basic terms associated with angles. 2. Find measures of complementary and supplementary angles. 3. Calculate with degrees, minutes, and seconds. 4. Convert between decimal degrees and degrees, minutes, and seconds. 5. Identify the characteristics of an angle in standard position. 6. Find measures of coterminal angles. 7. Find angle measures by using geometric properties. 8. Apply the angle sum of a triangle property. 2

3 Learning Objectives (Cont.) 9. Find angle measures and side lengths in similar triangles. 10. Solve applications involving similar triangles. 11. Learn basic concepts about trigonometric functions. 12. Find function values of an angle or quadrantal angles. 13. Decide whether a value is in the range of a trigonometric function 14. Use the reciprocal, Pythagorean and quotient identities. 15. Identify the quadrant of an angle. 16. Find other function values given one value and the quadrant. 3

4 Trigonometric Functions There are four major topics in this module: - Angles - Angle Relationships and Similar Triangles - Trigonometric Functions - Using the Definitions of the Trigonometric Functions 4

5 What are the basic terms? Two distinct points determine a line called line AB. A B Line segment AB a portion of the line between A and B, including points A and B. A Ray AB portion of line AB that starts at A and continues through B, and on past B. A B B 5

6 What are the basic terms? (cont.) 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. 6

7 How to Identify a Positive Angle and a Negative Angle? Positive angle: The rotation of the terminal side of an angle counterclockwise. Negative angle: The rotation of the terminal side is clockwise. 7

8 Most Common unit and Types of Angles The most common unit for measuring angles is the degree. The major types of angles are acute angle, right angle, obtuse angle and straight angle. 8

9 What are Complementary Angles? When the two angles form a right angle, they are complementary angles. Thus, we can find the measure of each angle in this case. k +20 k 16 The two angles have measures of = 63 and = 27 9

10 What are Supplementary Angles? When the two angles form a straight angle, they are supplementary angles. Thus, we can find the measure of each angle in this case too. 6x + 7 3x + 2 These angle measures are 6(19) + 7 = 121 and 3(19) + 2 = 59 10

11 How to Convert a Degree to Minute or Second? One minute is 1/60 of a degree. One second is 1/60 of a minute. 11

12 Example Perform the calculation. Perform the calculation. Write Since 86 = , the sum is written 12

13 Example Convert Convert

14 How to Determine an Angle is in 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. 14

15 What are 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. 15

16 What are 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. 16

17 Example 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. o o o a) (360 ) = 35 b) =

18 What are Vertical Angles? Vertical Angles have equal measures. Q R M N P The pair of angles NMP and RMQ are vertical angles. 18

19 Parallel Lines and Transversal 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. Transversal q m n parallel lines 19

20 Important Angle Relationships q m n Name Alternate interior angles Alternate exterior angles Interior angles on the same side of the transversal Corresponding angles Angles 4 and 5 3 and 6 1 and 8 2 and 7 4 and 6 3 and 5 2 & 6, 1 & 5, 3 & 7, 4 & 8 Rule Angles measures are equal. Angle measures are equal. Angle measures add to 180. Angle measures are equal. 20

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

22 Angle Sum of a Triangle The sum of the measures of the angles of any triangle is

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

24 Types of Triangles: Angles Note: The sum of the measures of the angles of any triangle is

25 Types of Triangles: Sides Again, the sum of the measures of the angles of any triangle is

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

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

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

Example of Application A lighthouse casts a

29 Example of Application 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. 4 3 x The lighthouse is 48 m high

30 The Six Trigonometric Functions Let (x, y) be a point other the origin on the terminal side of an angle θ in standard position. The distance from the point to the origin is The six trigonometric functions of θ are defined as follows. 30

31 Example of Finding Function Values The terminal side of angle θ in standard position passes through the point (12, 16). Find the values of the six trigonometric functions of angle θ. (12, 16) 16 θ 12 31

32 Example of Finding Function Values (cont.) Since x = 12, y = 16, and r = 20, we have 32

33 Another Example Find the six trigonometric function values of the angle θ in standard position, if the terminal side of θ is defined by x + 2y = 0, x 0. We can use any point on the terminal side of θ to find the trigonometric function values. 33

34 Choose x = 2 Another Example (cont.) Use the definitions: The point (2, 1) lies on the terminal side, and the corresponding value of r is 34

35 Example of Finding Function Values with 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. 35

36 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. 36

37 What are the Commonly Used Function Values? θ sin θ cos θ tan θ cot θ sec θ csc θ undefined 1 undefined undefined 0 undefined undefined 1 undefined undefined 0 undefined undefined 1 undefined 37

38 Reciprocal Identities 38

39 Example of Finding Function Values Using Reciprocal Identities Find cos θ if sec θ = Find sin θ if csc θ Since cos θ is the reciprocal of sec θ 39

40 Signs of Function Values at Different Quadrants θ in Quadrant sin θ cos θ tan θ cot θ sec θ csc θ I II + + III + + IV

41 Identify the Quadrant Identify the quadrant (or quadrants) of any angle θ that satisfies tan θ > 0 and cot θ > 0. tan θ > 0 in quadrants I and III cot θ > 0 in quadrants I and III so, the answer is quadrants I and III 41

42 Ranges of Trigonometric Functions For any angle θ for which the indicated functions exist: 1. 1 sin θ 1 and 1 cos θ 1; 2. tan θ and cot θ can equal any real number; 3. sec θ 1 or sec θ 1 and csc θ 1 or csc θ 1. (Notice that sec θ and csc θ are never between 1 and 1.) 42

43 Pythagorean Identities 43

44 Quotient Identities 44

45 Example of Other Function Values Find sin θ and cos θ if tan θ = 4/3 and θ is in quadrant III. Since θ is in quadrant III, sin θ and cos θ will both be negative. sin θ and cos θ must be in the interval [ 1, 1]. 45

46 Example of Other Function Values (cont.) We use the identity 46

47 We have learned to What have we learned? 1. Use basic terms associated with angles. 2. Find measures of complementary and supplementary angles. 3. Calculate with degrees, minutes, and seconds. 4. Convert between decimal degrees and degrees, minutes, and seconds. 5. Identify the characteristics of an angle in standard position. 6. Find measures of coterminal angles. 7. Find angle measures by using geometric properties. 8. Apply the angle sum of a triangle property. 47

48 What have we learned? (Cont.) 9. Find angle measures and side lengths in similar triangles. 10. Solve applications involving similar triangles. 11. Learn basic concepts about trigonometric functions. 12. Find function values of an angle or quadrantal angles. 13. Decide whether a value is in the range of a trigonometric function 14. Use the reciprocal, Pythagorean and quotient identities. 15. Identify the quadrant of an angle. 16. Find other function values given one value and the quadrant. 48

49 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition 49

MAC Learning Objectives. Learning Objectives (Cont.) Module 2 Acute Angles and Right Triangles

MAC Learning Objectives. Learning Objectives (Cont.) Module 2 Acute Angles and Right Triangles MAC 1114 Module 2 Acute Angles and Right Triangles Learning Objectives Upon completing this module, you should be able to: 1. Express the trigonometric ratios in terms of the sides of the triangle given