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Pre-Class Problems 7 for Wednesday, February 13 Illustration of the definition of the cosine function, sine function, tangent function, secant function, cosecant function, and cotangent function for the acute angle using right triangle trigonometry. Second illustration of the cosine function, sine function, tangent function, secant function, cosecant function, and cotangent function. Illustration of the definition of all the six trigonometric functions for the acute angle using right triangle trigonometry. Second illustration of all the six trigonometric functions. These are the type of problems that you will be working on in class. These problems are from Lesson 6. Solution to Problems on the Pre-Exam. You can go to the solution for each problem by clicking on the problem letter. Objective of the following problems: To use right triangle trigonometry to find the exact value of the six trigonometric functions for an acute angle. You will also need to apply the Pythagorean Theorem. 1. Find the exact value of cosine, sine, and tangent for the following angles. a. b. 11 7 5 6 c. 8 4 Objective of the following problems: To use right triangle trigonometry to find the exact value of the other five trigonometric functions for an acute angle if given the value of one of the trigonometric function of the angle. 2. Use a right triangle to find the exact value of the other five trigonometric functions of the angle if given the following information. a. cos b. tan c. csc 4 and is an acute angle 7 3 and is an acute angle 5 8 and is an acute angle 3 Objective of the following problems: To determine what quadrant the terminal side of an angle is in if given the sign of two trigonometric functions of the angle. 3. Determine what quadrant the following angles are in if given the following information. a. cos 0 and sin 0 b. sin 0 and sec 0 c. cos 0 and csc 0 d. sin 0 and tan 0 e. cos 0 and cot 0 f. tan 0 and sec 0 g. cot 0 and csc 0 h. csc 0 and sec 0 Additional problems available in the textbook: Page 492 … 13 – 18, 25 - 30. Page 483 … Examples1 and 2. Page 505 … 9 – 14. Solutions: 1a. 5 7 24 NOTE: You use the Pythagorean Theorem to find the value of cos adj hyp 24 , 7 Answer: cos Back to Problem 1. 24 , 7 sin opp 5 , hyp 7 sin 5 , 7 tan tan 5 24 opp adj 24 . 5 24 11 1b. 5 6 NOTE: You use the Pythagorean Theorem to find the value of 5. cos 11 , 6 adj hyp Answer: cos sin 11 , 6 opp 5 , hyp 6 sin 5 , 6 tan tan opp adj 5 11 5 11 Back to Problem 1. 1c. 80 4 5 8 4 NOTE: You use the Pythagorean Theorem to find the value of cos adj 8 hyp 4 5 2 , 5 sin opp 4 hyp 4 5 1 , 5 80 . tan opp 4 1 adj 8 2 2 , 5 Answer: cos sin 1 , 5 tan 1 2 Back to Problem 1. 2a. cos 4 and is an acute angle 7 cos 4 7 sec 7 4 NOTE: We do not need to use right triangle trigonometry to find this value. cos NOTE: 4 adj 7 hyp 7 33 4 NOTE: You use the Pythagorean Theorem to find the value of sin opp hyp 33 7 csc hyp opp 7 33 33 . tan opp adj 33 4 Answer: sin cot cot 33 , 7 33 , 4 tan adj opp sec 4 33 7 , 4 csc 7 , 33 4 33 Back to Problem 2. 2b. tan 3 and is an acute angle 5 tan 3 5 cot 5 3 NOTE: We do not need to use right triangle trigonometry to find this value. tan NOTE: 3 opp 5 adj 28 3 5 NOTE: You use the Pythagorean Theorem to find the value of 28 . cos adj hyp 5 28 sec hyp adj 28 5 sin opp hyp 3 csc hyp opp 28 28 Answer: cos csc 5 , 28 sin 28 cot , 3 3 28 , sec 3 28 , 5 5 3 Back to Problem 2. 2c. csc 8 and is an acute angle 3 csc 8 3 sin 3 8 NOTE: We do not need to use right triangle trigonometry to find this value. sin NOTE: 3 opp 8 hyp 8 3 55 NOTE: You use the Pythagorean Theorem to find the value of cos adj hyp 55 8 sec hyp adj 8 55 tan opp adj 3 55 cot adj opp 55 3 Answer: cos 55 , 8 sin sec 8 , 55 cot 3 , 8 tan 55 . 3 , 55 55 3 Back to Problem 2. 3a. cos 0 and sin 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) cos 0 x 0 sin 0 y 0 Answer: IV NOTE: The quadrant where x-coordinates are positive and y-coordinates are negative is the fourth quadrant. Using Method 2 from Lesson 6: sin 0 y cos 0 y X x X x X X Answer: IV Back to Problem 3. 3b. sin 0 and sec 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) sin 0 y 0 sec 0 cos 0 x 0 Answer: II NOTE: The quadrant where x-coordinates are negative and y-coordinates are positive is the second quadrant. NOTE: Cosine and secant are reciprocals of each other. The sign of reciprocals is the same. Using Method 2 from Lesson 6: sec 0 or cos 0 y sin 0 y X X X x x X Answer: II Back to Problem 3. 3c. cos 0 and csc 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) cos 0 x 0 csc 0 sin 0 y 0 Answer: III NOTE: The quadrant where x-coordinates are negative and y-coordinates are negative is the third quadrant. NOTE: Sine and cosecant are reciprocals of each other. The sign of reciprocals is the same. Using Method 2 from Lesson 6: csc 0 or sin 0 y cos 0 y X x X x X X Answer: III Back to Problem 3. 3d. sin 0 and tan 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) sin 0 y 0 tan 0 : ( ) tan y () x 0 x ? Answer: IV NOTE: The quadrant where x-coordinates are positive and y-coordinates are negative is the fourth quadrant. NOTE: A negative divided by a positive will produce a negative result. Division with unlike signs will produce a negative. Using Method 2 from Lesson 6: tan 0 y sin 0 y X x X x X X Answer: IV Back to Problem 3. 3e. cos 0 and cot 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) cos 0 x 0 cot 0 tan 0 : ( ) tan y ? y 0 x () Answer: III NOTE: The quadrant where x-coordinates are negative and y-coordinates are negative is the third quadrant. NOTE: Tangent and cotangent are reciprocals of each other. The sign of reciprocals is the same. NOTE: A negative divided by a negative will produce a positive result. Division with like signs will produce a positive. Using Method 2 from Lesson 6: cot 0 or tan 0 y cos 0 y X X x x X X Answer: III Back to Problem 3. 3f. tan 0 and sec 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) sec 0 cos 0 x 0 tan 0 : ( ) tan y ? y 0 x () Answer: II NOTE: The quadrant where x-coordinates are negative and y-coordinates are positive is the second quadrant. NOTE: Cosine and secant are reciprocals of each other. The sign of reciprocals is the same. NOTE: A positive divided by a negative will produce a negative result. Division with unlike signs will produce a negative. Using Method 2 from Lesson 6: sec 0 or cos 0 y tan 0 y X X x x X X Answer: II Back to Problem 3. 3g. cot 0 and csc 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) csc 0 sin 0 y 0 cot 0 tan 0 : ( ) tan y () x 0 x ? Answer: II NOTE: The quadrant where x-coordinates are negative and y-coordinates are positive is the second quadrant. NOTE: Sine and cosecant are reciprocals of each other. The sign of reciprocals is the same. NOTE: Tangent and cotangent are reciprocals of each other. The sign of reciprocals is the same. NOTE: A positive divided by a negative will produce a negative result. Division with unlike signs will produce a negative. Using Method 2 from Lesson 6: cot 0 or tan 0 y X csc 0 or sin 0 y X x X x X Answer: II Back to Problem 3. 3h. csc 0 and sec 0 Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) csc 0 sin 0 y 0 sec 0 cos 0 x 0 Answer: IV NOTE: The quadrant where x-coordinates are positive and y-coordinates are negative is the fourth quadrant. NOTE: Sine and cosecant are reciprocals of each other. The sign of reciprocals is the same. NOTE: Cosine and secant are reciprocals of each other. The sign of reciprocals is the same. Using Method 2 from Lesson 6: sec 0 or cos 0 y csc 0 or sin 0 y X x X x X X Answer: IV Back to Problem 3. Solution to Problems on the Pre-Exam: 11. 7 Given: Back to Page 1. Find the exact value of a. cos 3 3 4 4 b. csc 4 7 8. If tan 0 and sec 0 , then lies in which quadrant? Using Method 1, which is Unit Circle Trigonometry, from Lesson 6: Recall Unit Circle Trigonometry: P ( ) ( cos , sin ) sec 0 cos 0 x 0 ( ) tan tan 0 : y ? y 0 x () Answer: II NOTE: The quadrant where x-coordinates are negative and y-coordinates are positive is the second quadrant. NOTE: Cosine and secant are reciprocals of each other. The sign of reciprocals is the same. NOTE: A positive divided by a negative will produce a negative result. Division with unlike signs will produce a negative. Using Method 2 from Lesson 6: sec 0 or cos 0 y tan 0 y X X x X Answer: II x X