Rev.S08 MAC 1114 Module 1 Trigonometric Functions

2 Rev.S08 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. Click link to download other modules.

3 Rev.S08 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. Click link to download other modules.

4 Rev.S08 Trigonometric Functions Click link to download other modules. - Angles - Angle Relationships and Similar Triangles - Trigonometric Functions - Using the Definitions of the Trigonometric Functions There are four major topics in this module:

5 Rev.S08 What are the basic terms? Click link to download other modules. 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. A B AB A B

6 Rev.S08 What are the basic terms? (cont.) Click link to download other modules. 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.

7 Rev.S08 How to Identify a Positive Angle and a Negative Angle? Click link to download other modules. Positive angle: The rotation of the terminal side of an angle counterclockwise. Negative angle: The rotation of the terminal side is clockwise.

8 Rev.S08 Most Common unit and Types of Angles Click link to download other modules. 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.

9 Rev.S08 What are Complementary Angles? Click link to download other modules. 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 − 16 k +20 The two angles have measures of = 63  and 43 − 16 = 27 

10 Rev.S08 What are Supplementary Angles? Click link to download other modules. 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 + 73x + 2 These angle measures are 6(19) + 7 = 121  and 3(19) + 2 = 59 

11 Rev.S08 How to Convert a Degree to Minute or Second? Click link to download other modules. One minute is 1/60 of a degree. One second is 1/60 of a minute.

12 Rev.S08 Example Click link to download other modules. Perform the calculation. Since 86 = , the sum is written Perform the calculation. Write

13 Rev.S08 Example Click link to download other modules. Convert Convert 

14 Rev.S08 How to Determine an Angle is in Standard Position? Click link to download other modules. An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis.

15 Rev.S08 What are Quadrantal Angles? Click link to download other modules. 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 Rev.S08 What are Coterminal Angles? Click link to download other modules. 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.

17 Rev.S08 Example Click link to download other modules. 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   = 173  (360 ) −  35   

18 Rev.S08 What are Vertical Angles? Click link to download other modules. Vertical Angles have equal measures. The pair of angles NMP and RMQ are vertical angles. M Q R P N

19 Rev.S08 Parallel Lines and Transversal Click link to download other modules. 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. m n parallel lines q Transversal

20 Rev.S08 Important Angle Relationships Click link to download other modules. Angle measures are equal.2 & 6, 1 & 5, 3 & 7, 4 & 8 Corresponding angles Angle measures add to 180 . 4 and 6 3 and 5 Interior angles on the same side of the transversal Angle measures are equal.1 and 8 2 and 7 Alternate exterior angles Angles measures are equal.4 and 5 3 and 6 Alternate interior angles RuleAnglesName m n q

21 Rev.S08 Example of Finding Angle Measures Click link to download other modules. 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. m n (10x − 80)  (6x + 4)  One angle has measure 6x + 4 = 6(21) + 4 = 130  and the other has measure 10x − 80 = 10(21) − 80 = 130 

22 Rev.S08 Angle Sum of a Triangle Click link to download other modules. The sum of the measures of the angles of any triangle is 180 .

23 Rev.S08 Example of Applying the Angle Sum Click link to download other modules. 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 . 52  65  xx

24 Rev.S08 Types of Triangles: Angles Click link to download other modules. Note: The sum of the measures of the angles of any triangle is 180 .

25 Rev.S08 Types of Triangles: Sides Click link to download other modules. Again, the sum of the measures of the angles of any triangle is 180 .

26 Rev.S08 What are the Conditions for Similar Triangles? Click link to download other modules. Corresponding angles must have the same measure. Corresponding sides must be proportional. (That is, their ratios must be equal.)

27 Rev.S08 Example of Finding Angle Measures Click link to download other modules. 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  A CB F E D 35  112  33  112 

28 Rev.S08 Example of Finding Side Lengths Click link to download other modules. Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF. A CB F E D 35  112  33  112  To find side DE. To find side FE.

29 Rev.S08 Example of Application Click link to download other modules. The two triangles are similar, so corresponding sides are in proportion. The lighthouse is 48 m high. 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 x

30 Rev.S08 The Six Trigonometric Functions Click link to download other modules. 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.

31 Rev.S08 Example of Finding Function Values Click link to download other modules. The terminal side of angle  in standard position passes through the point (12, 16). Find the values of the six trigonometric functions of angle . (12, 16) 

32 Rev.S08 Example of Finding Function Values (cont.) Click link to download other modules. Since x = 12, y = 16, and r = 20, we have

33 Rev.S08 Another Example Click link to download other modules. Find the six trigonometric function values of the angle  in standard position, if the terminal side of  is defined by x + 2y = 0, x  0. We can use any point on the terminal side of  to find the trigonometric function values.

34 Rev.S08 Another Example (cont.) Click link to download other modules. Choose x = 2 The point (2, −1) lies on the terminal side, and the corresponding value of r is Use the definitions:

35 Rev.S08 Example of Finding Function Values with Quadrantal Angles Click link to download other modules. 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.

36 Rev.S08 Undefined Function Values Click link to download other modules. 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.

37 Rev.S08 What are the Commonly Used Function Values? Click link to download other modules. undefined  −1−1undefined0 0−1−1 270  undefined−1−1 0−1−  1undefined  undefined 0 csc  sec  cot  tan  cos  sin 

38 Rev.S08 Reciprocal Identities Click link to download other modules.

39 Rev.S08 Example of Finding Function Values Using Reciprocal Identities Click link to download other modules. Find cos  if sec  = Since cos  is the reciprocal of sec  Find sin  if csc 

40 Rev.S08 Signs of Function Values at Different Quadrants Click link to download other modules. −+−−+− IV −−++−− III +−−−−+ II I csc  sec  cot  tan  cos  sin  in Quadrant

41 Rev.S08 Identify the Quadrant Click link to download other modules. Identify the quadrant (or quadrants) of any angle  that satisfies tan  > 0 and cot  > 0. tan  > 0 in quadrants I and III cot  > 0 in quadrants I and III so, the answer is quadrants I and III

42 Rev.S08 Ranges of Trigonometric Functions Click link to download other modules. 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.)

43 Rev.S08 Pythagorean Identities Click link to download other modules.

44 Rev.S08 Quotient Identities Click link to download other modules.

45 Rev.S08 Example of Other Function Values Click link to download other modules. Find sin  and cos  if tan  = 4/3 and  is in quadrant III. Since  is in quadrant III, sin  and cos  will both be negative. sin  and cos  must be in the interval [−1, 1].

46 Rev.S08 Example of Other Function Values (cont.) Click link to download other modules. We use the identity

47 Rev.S08 What have we learned? We have learned 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. Click link to download other modules.

48 Rev.S08 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. Click link to download other modules.

49 Rev.S08 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 Click link to download other modules.