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Pre-Class Problems 2 for Thursday, September 11 These are the type of problems that you will be working on in class. These problems are from Lesson 2 and Lesson 3. Solution to Problems on the Pre-Exam. You can go to the solution for each problem by clicking on the problem letter. 1. Find the exact value of the six trigonometric functions (cosine, sine, tangent, secant, cosecant, and cotangent) of the following angles. NOTE: The objective of these problems is to use Unit Circle Trigonometry to find the value of any of the six trigonometry functions of the angles whose terminal sides are located on one of the coordinate axes. 2. 3 2 a. 0 b. 180 c. d. 270 e. 90 f. g. 2 h. 360 i. 2 Find the exact value of the six trigonometric functions (cosine, sine, tangent, secant, cosecant, and cotangent) of the following angles. NOTE: The objective of these problems is to use either Unit Circle Trigonometry or Right Triangle Trigonometry to find the value of any of the six trigonometry functions of these three acute angles. a. 3. 30 b. 3 c. 4 For each angle, determine the location of the angle and then find the reference angle of the angle if it has one. Show your work for finding the reference angle. NOTE: The objective of these problems is to learn how to find the reference angle of an angle if it has one. The reference angle of an angle is an acute angle. a. 210 d. g. j. 270 b. 325 2 3 e. 3 8 h. 135 k. 6 17 45 26 c. f. i. 15 11 2 Additional problems available in the textbook: Page 137 … 9, 10, 13, 14, 21, 22, 28, 29, 30. Page 134 … Examples 1a and 1b. Page 140 … Examples 2 and 3. Page 157 … 45 - 50. Page 152 … Example 4. Solutions: 1a. 0 Animation of the making of the 0 angle. NOTE: The angle of 0 radians is the same as the angle of 0 . The terminal side of this angle is the positive x-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( 0 ) P ( 0 ) ( 1, 0 ) cos 0 x 1 sec 0 1 1 1 x 1 sin 0 y 0 csc 0 1 1 undefined y 0 tan 0 y 0 0 x 1 cot 0 x 1 undefined y 0 Back to Problem 1. 1b. 180 Animation of the making of the 180 angle. NOTE: The angle of 180 is the same as the angle of radians. The terminal side of this angle is the negative x-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P (180 ) P ( ) ( 1, 0 ) cos 180 x 1 sec 180 1 1 1 x 1 sin 180 y 0 csc 180 1 1 undefined y 0 cot 180 x 1 undefined y 0 tan 180 y 0 0 x 1 Back to Problem 1. 1c. 3 2 Animation of the making of the 3 angle. 2 3 radians is the same as the angle of 270 . The 2 terminal side of this angle is the positive y-axis. NOTE: The angle of NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) 3 P P ( 270 ) ( 0 , 1) 2 3 cos x 0 2 1 1 3 sec undefined x 0 2 3 sin y 1 2 1 1 3 csc 1 y 1 2 y 1 3 tan undefined x 0 2 x 0 3 cot 0 y 1 2 Back to Problem 1. 1d. 270 Animation of the making of the 270 angle. NOTE: The angle of 270 is the same as the angle of 3 radians. The 2 terminal side of this angle is the negative y-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( 270 ) P ( ) ( 0 , 1) cos 270 x 0 sin 270 y 1 tan 270 y 1 undefined x 0 sec 270 1 1 undefined x 0 csc 270 1 1 1 y 1 cot 270 x 0 0 y 1 Back to Problem 1. 1e. 90 Animation of the making of the 90 angle. NOTE: The angle of 90 is the same as the angle of radians. The terminal 2 side of this angle is the positive y-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( 90 ) P ( 0 , 1 ) 2 cos 90 x 0 sec 90 1 1 undefined x 0 sin 90 y 1 csc 90 1 1 1 y 1 cot 90 x 0 0 y 1 tan 90 y 1 undefined x 0 Back to Problem 1. 1f. Animation of the making of the angle. NOTE: The angle of radians is the same as the angle of 180 . The terminal side of this angle is the negative x-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( ) P ( 180 ) ( 1, 0 ) cos ( ) x 1 sec ( ) 1 1 1 x 1 sin ( ) y 0 csc ( ) 1 1 undefined y 0 tan ( ) y 0 0 x 1 cot ( ) x 1 undefined y 0 Back to Problem 1. 1g. 2 NOTE: The angle of Animation of the making of the angle. 2 radians is the same as the angle of 90 . The 2 terminal side of this angle is the negative y-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P P ( 90 ) ( 0 , 1 ) 2 cos x 0 2 1 1 sec undefined x 0 2 sin y 1 2 y 1 tan undefined 2 x 0 Back to Problem 1. 1 1 csc 1 y 1 2 x 0 cot 0 2 y 1 1h. 360 Animation of the making of the 360 angle. NOTE: The angle of 360 is the same as the angle of 2 radians. The terminal side of this angle is the positive x-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( 360 ) P ( 2 ) ( 1, 0 ) cos ( 360 ) x 1 sec ( 360 ) 1 1 1 x 1 sin ( 360 ) y 0 csc ( 360 ) 1 1 undefined y 0 cot ( 360 ) x 1 undefined y 0 tan ( 360 ) y 0 0 x 1 Back to Problem 1. 1i. 2 Animation of the making of the 2 angle. NOTE: The angle of 2 is the same as the angle of 360 . The terminal side of this angle is the positive x-axis. NOTE: Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( 2 ) P ( 360 ) ( 1, 0 ) cos 2 x 1 sec 2 1 1 1 x 1 sin 2 y 0 csc 2 1 1 undefined y 0 tan 2 y 0 0 x 1 cot 2 x 1 undefined y 0 Back to Problem 1. 2a. 30 Animation of the making of the 30 angle. NOTE: The angle of 30 is the same as the angle of radians. The terminal 6 side of this angle is in the I quadrant. Using Unit Circle Trigonometry: P ( ) ( cos , sin ) P ( 30 ) P 6 3 2 cos 30 x sin 30 y tan 30 y x 3 1 , 2 2 1 2 1 2 3 2 1 3 Using Right Triangle Trigonometry: sec 30 1 x csc 30 1 2 y cot 30 x y 2 3 3 2 1 2 3 60 2 1 30 3 3 2 sec 30 hyp adj opp 1 hyp 2 csc 30 hyp 2 2 opp 1 opp adj cot 30 adj opp cos 30 adj hyp sin 30 tan 30 1 3 2 3 3 1 3 Back to Problem 2. 2b. 3 Animation of the making of the angle. 3 NOTE: The angle of radians is the same as the angle of 60 . The terminal 3 side of this angle is in the I quadrant. Using Unit Circle Trigonometry: P ( ) ( cos , sin ) 1 P P ( 60 ) , 2 3 3 2 cos 1 x 3 2 sin y 3 3 2 y 3 x 3 2 1 2 tan sec 1 2 3 x csc 1 2 3 y 3 cot 3 x 3 y 1 2 3 2 1 3 Using Right Triangle Trigonometry: 60 2 1 30 3 cos adj 1 3 hyp 2 sec hyp 2 2 3 adj 1 sin opp 3 hyp 3 2 csc hyp 2 3 opp 3 tan opp 3 adj 3 1 cot adj 1 3 opp 3 Back to Problem 2. 3 2c. 4 Animation of the making of the NOTE: The angle of angle. 4 radians is the same as the angle of 45 . The terminal 4 side of this angle is in the I quadrant. Using Unit Circle Trigonometry: P ( ) ( cos , sin ) P P ( 45 ) 4 2 , 2 2 2 cos x 4 2 2 sec 1 4 x 2 2 2 sin y 4 2 2 csc 1 4 y 2 2 2 x 4 y 2 2 1 2 2 tan y 4 x 2 2 1 2 2 cot Using Right Triangle Trigonometry: 45 2 1 45 1 cos adj 4 hyp 1 2 sec hyp 4 adj 2 1 2 sin opp 4 hyp 1 2 csc hyp 4 opp 2 1 2 tan opp 1 1 4 adj 1 cot adj 1 1 4 opp 1 Back to Problem 2. 3a. 210 Animation of the making of the 210 angle. 210 180 and 210 270 is in III 180 ' 210 ' 30 210 ' OR ' 210 180 30 Answer: 30 Back to Problem 3. 3b. 325 Animation of the making of the 325 angle. 325 180 and 325 270 is in I 325 ' 360 ' 35 325 ' OR ' 360 325 35 Answer: 35 Back to Problem 3. 3c. 45 26 Animation of the making of the 45 angle. 26 45 26 45 3 39 is in IV and 26 26 26 2 26 45 7 ' 2 ' 26 26 45 7 ' 2 OR 26 26 45 26 ' Answer: 7 26 Back to Problem 3. 3d. 2 3 Animation of the making of the 2 angle. 3 2 3 2 1.5 is in II and 3 3 3 2 3 2 ' ' 3 3 ' 2 3 OR ' Answer: 2 3 3 3 Back to Problem 3. 3e. 6 17 Animation of the making of the 6 angle. 17 6 17 6 8.5 is in IV and 17 17 17 2 17 6 17 ' 0 ' 6 6 ' 17 17 OR ' 6 6 0 17 17 Answer: 3f. 6 17 Back to Problem 3. 15 11 Animation of the making of the 15 angle. 11 15 11 15 3 16.5 is in II and 11 11 11 2 11 ' ' 15 4 ' 11 11 OR ' 15 4 11 11 15 11 Answer: 4 11 Back to Problem 3. 3g. 3 8 Animation of the making of the 3 8 3 4 is in I and 8 8 8 2 8 3 angle. 8 3 8 ' 0 ' 3 3 ' 8 8 OR ' 3 3 0 8 8 Answer: 3 8 Back to Problem 3. 3h. 135 Animation of the making of the 135 angle. 135 180 and 135 90 is in III 135 ' 180 ' 45 OR ' 180 135 45 ' 135 Answer: 45 Back to Problem 3. 3i. 2 Animation of the making of the angle. 2 Answer: No reference angle 2 Back to Problem 3. 3j. 270 Animation of the making of the 270 angle. Answer: No reference angle 270 Back to Problem 3. 3k. Animation of the making of the angle. Answer: No reference angle Back to Problem 3. Solution to Problems on the Pre-Exam: 1b. tan 180 Back to Page 1. Animation of the making of the 180 angle. P (180 ) ( 1, 0 ) tan 180 y 0 0 x 1 Answer: 0 1a. Reference angle of 4 3 Animation of the making of the 4 angle. 3 4 3 4 3 4.5 is in III and 3 3 3 2 3 4 ' 3 3 4 ' OR 3 3 ' 4 3 ' Answer: ' 3 1b. Reference angle of 180 Animation of the making of the 180 angle. Answer: ' No reference angle 180 1c. Reference angle of 315 Animation of the making of the 315 angle. 315 180 and 315 270 is in I 315 ' 360 ' 45 315 ' OR ' 360 315 45 Answer: ' 45 5. Reference angle of 290 Animation of the making of the 290 angle. 290 180 and 290 270 is in IV 290 ' 360 ' 70 OR ' 360 290 70 290 ' Answer: ' 70 6. Reference angle of 11 14 Animation of the making of the 11 angle. 14 11 14 11 7 is in III and 14 14 14 2 14 11 3 ' ' 14 14 OR ' ' 11 14 11 3 14 14 Answer: ' 3 14