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MAC 1114 Module 1 Trigonometric Functions Rev.S08 Learning Objectives • Upon completing this module, you should be able to: 1. 2. 3. 4. Use basic terms associated with angles. Find measures of complementary and supplementary angles. Calculate with degrees, minutes, and seconds. Convert between decimal degrees and degrees, minutes, and seconds. Identify the characteristics of an angle in standard position. Find measures of coterminal angles. Find angle measures by using geometric properties. Apply the angle sum of a triangle property. 5. 6. 7. 8. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 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 B Ray AB—portion of line AB that starts at A and continues through B, and on past B. A Rev.S08 B http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 How to Identify a Positive Angle and a Negative Angle? Positive angle: The rotation of the terminal side of an angle counterclockwise. Rev.S08 Negative angle: The rotation of the terminal side is clockwise. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 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 43 + 20 = 63 and 43 16 = 27 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 How to Convert a Degree to Minute or Second? Rev.S08 One minute is 1/60 of a degree. One second is 1/60 of a minute. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 11 Example Perform the calculation. Perform the calculation. Write Since 86 = 60 + 26, the sum is written Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 Example Convert Rev.S08 Convert 36.624 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 13 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 14 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 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) 1115 3(360 ) = 35 b) 187 + 360 = 173 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 What are Vertical Angles? Vertical Angles have equal measures. Q R M N P The pair of angles NMP and RMQ are vertical angles. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 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 parallel lines n Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 Important Angle Relationships 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, Angle measures are equal. 3 & 7, 4 & 8 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 Example of Finding Angle Measures Find the measure of each marked angle, given that lines m and n are parallel. (6x + 4) m (10x 80) n The marked angles are alternate exterior angles, which are equal. Rev.S08 One angle has measure 6x + 4 = 6(21) + 4 = 130 and the other has measure 10x 80 = 10(21) 80 = 130 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 Angle Sum of a Triangle The sum of the measures of the angles of any triangle is 180. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 22 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. 65 Solution The third angle of the triangle measures 63. x 52 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 23 Types of Triangles: Angles Note: The sum of the measures of the angles of any triangle is 180. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 Types of Triangles: Sides Again, the sum of the measures of the angles of any triangle is 180. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 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.) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 26 Example of Finding Angle Measures Triangles ABC and DEF are similar. Find the measures of angles D and E. D Since the triangles are similar, corresponding angles have the same measure. Angle D corresponds to angle A which = 35 A 112 35 F C Rev.S08 112 33 E Angle E corresponds to angle B which = 33 B http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 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. To find side FE. D A 16 112 35 64 F E 32 C 112 33 B 48 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 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. The lighthouse is 48 m high. 3 4 x 64 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 29 The Six 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 Example of 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 . (12, 16) 16 12 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 31 Example of Finding Function Values (cont.) Since x = 12, y = 16, and r = 20, we have Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 32 Another Example 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 33 Another Example (cont.) Choose x = 2 Use the definitions: The point (2, 1) lies on the terminal side, and the corresponding value of r is Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 34 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 35 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 36 What are the Commonly Used Function Values? 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 37 Reciprocal Identities Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 38 Example of Finding Function Values Using Reciprocal Identities Find cos if sec = Find sin if csc Since cos is the reciprocal of sec Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 39 Signs of Function Values at Different Quadrants in sin Quadrant cos tan cot sec csc I + + + + + + II + + III + + IV + + Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 40 Identify the Quadrant 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 41 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.) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 42 Pythagorean Identities Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 43 Quotient Identities Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 44 Example of Other Function Values 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]. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 45 Example of Other Function Values (cont.) We use the identity Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 46 What have we learned? • We have learned to 1. 2. 3. 4. Use basic terms associated with angles. Find measures of complementary and supplementary angles. Calculate with degrees, minutes, and seconds. Convert between decimal degrees and degrees, minutes, and seconds. Identify the characteristics of an angle in standard position. Find measures of coterminal angles. Find angle measures by using geometric properties. Apply the angle sum of a triangle property. 5. 6. 7. 8. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 47 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 48 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 49