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Trigonometry Section 1.1 Graph the points on a coordinate system and identify the quadrant or axis for each point. 1. A 3, 2 2. B 8, 3 3. C 7, 1 4. D 5,1 5. E 0, 4 6. F 6,0 Use the distance formula to find the distance between the following pairs of points. 7. W 2,1 and Z 3, 4 8. X 5, 2 and Y 3, 7 A triple of positive integers a, b, c is called a Pythagorean Triple if it satisfies the equation of the Pythagorean Theorem. Determine whether each of the following is a Pythagorean Triple. 9. 9,12,15 10. 6,8,10 5,10,15 11. Using both the distance formula and the Pythagorean Theorem, determine if the triangle made up of the following points is a right triangle or not. 12. J 2,5 , K 1,5 and L 1,9 Find all values of 13. x such that the distance between the given points is as indicated. x,7 and 2,3 is 5 14. 5, x and 8,1 is 5 Use the midpoint formula to find the midpoint of the line segment joining the two points. 15. 1,3 and 7,5 16. 17. A line segment has an endpoint at endpoint. 18. A line segment has an endpoint at endpoint. 4,3 and 1,2 3,2 and midpoint 5,3 . Find the other missing 4,1 and midpoint 5,6 . Find the other missing Change the following set notations to interval notations. 19. Let 22. x x 6 20. y y 9 21. x 3 x 6 f x 2 x 2 4 x 6 . Find each of the following. f 2 23. f 3 24. f a 25. f 2 p Find the domain and range of the following equations or graphs. Identify any which are functions. 26. 2 x 5 y 10 27. y 2x2 5 28. y 4 x 29. x y2 30. 31. Find the domain only of the following equations. 32. y 1 x 33. y 34. y 2 x 1 35. y x 4 x2 5x 13x 8 Trigonometry Section 1.2 Find the measure of each angle. 1. 3. 2y 4y 2. 5k 5 3k 5 Supplementary angles with measures 6x 4 and 8x 12 degrees. Perform each calculation. 4. 6218 2141 7. 180 12451 5. 7515 8332 6. 90 5128 Convert each angle measure to decimal degrees. (Round to the nearest thousandth of a degree) 8. 2054 9. 913554 10. 2741859 13. 178.5994 Convert each angle measure to degrees, minutes, and seconds. 11. 59.0854 12. 102.3771 Find the angles of smallest positive measure coterminal with the following angles. 14. 40 15. 125 16. 539 17. 850 Sketch the angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle. 18. 89 19. 174 20. 250 22. 52 23. 159 24. 438 21. 512 Locate the following points in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having the smallest positive measure. Then find the distance r from the origin to the point, using the distance formula. 25. 3,3 26. 5,2 27. 3,1 28. 4 3 ,4 Solve each problem. 29. A windmill makes 90 revolutions per minute. How many revolutions does it make per second? 30. A tire is rotating 600 times per minute. Through how many degrees does a point on the edge of the tire move in 1 second? 2 Trigonometry Section 1.3 Use the properties of angle measures given in this section to find the measure of each marked angle. 1. 5x 129 2 x 21 2. a 2 x 5 a b x 22 b m 3. 4. x 1 n a 8 6 120 9 7 10 a b 4 x 56 4 b m n 1 5. 55 5 2 3 2x 120 6. x 20 x 210 3x 7. x 30 Find the measure of the third angle if the other two angles measure: a) 13650 and 413815 b) 29.6 and 49.7 c) 59.8 and 100.3 d) 4413 and 584 1 x 15 2 Classify each triangle as acute, right, equiangular or obtuse and also equilateral, isosceles or scalene. 8. 130 9. 8 10. 8 r r r 11. 12. 10 96 65 65 14 Find all unknown angle measures or sides in each pair of similar triangles. 42 13. A Q 14. N M B 106 K C B R P A C 30 C 12 85 15. 16. 24 I P 45 J 10 A 8 F 7 G 14 N K M 21 B 10.5 H 8 17. N 18. 6 Y B Q 5 25 A 9 O M 35 4 Z P C 10 R X Find the value of the variables. 19. 20. Solve each problem. 21. A forest fire lookout tower casts a shadow 180 feet long at the same time that the shadow of a 9 foot truck is 15 feet long. Find the height of the tower. 22. On a photograph of a triangular piece of land, the lengths of the three sides are 4cm, 5 cm and 7 cm, respectively. The shortest side of the actual piece of land is 450 m long. Find the lengths of the other two sides. 23. Sam built a ramp to a loading dock. The ramp has a vertical support of 2 meters from the base of the loading dock and 3 meters form the base of the ramp. If the vertical support is 1.2 meters in height, what is the height of the loading dock? 24. A tourist that is 192 cm tall wants to estimate the height of an office tower. He places a mirror on the ground 87.6 meters from the base of the tower and moves back 0.4 meters to sight the top of the tower in the mirror. How tall is the tower? 25. Lewis, an adventurous explorer, is trying to determine how wide a river is that he needs to cross. To do this he first calls the point where he is standing point A. Next, he locates a tree on the opposite side of the river that is directly across from where he is standing, which he calls point B. He then walks 32 paces along the river and marks this next spot as point C. He then walks 10 more paces along the river and marks this next spot as point D, where he turns and walks directly away from the river until point C lines up with point B. This takes 8 paces and he marks this final spot as point E. What is the width of the river in paces? Trigonometry Section 1.4 Find the values of the six trigonometric functions for the angles in standard position having the following points on their terminal sides. Rationalize the denominators when applicable. 1. 3,4 2. 12,5 3. 6,8 4. 24,7 5. 0,2 6. 4,0 7. 1, 3 8. 2 9. 2 10. 2 , 2 2 5 ,2 3 ,2 An equation with a restriction on x is given. This is an equation of the terminal side of an angle in standard position. Sketch the smallest positive such angle , and find the values of the six trigonometric functions of . 11. 4 x 7 y 0, x 0 12. 6 x 5 y 0, x 0 Use the trigonometric functions values of quadrantal angles given in this section to evaluate each of the following. An expression such as cot 90 means cot 90 which is equal to 0 0 . 2 2 2 13. tan 0 6 sin 90 14. 4 csc 270 3 cos180 15. 2 sec 0 4 cot 2 90 cos 360 16. sin 2 360 cos 2 360 17. sec 2 180 3 sin 2 360 2 cos180 18. 3 csc 2 270 2 sin 2 270 3 sin 270 Trigonometry Section 1.5 Use the appropriate reciprocal identity to find each function value. Rationalize the denominator when applicable. 1. sin if csc 3 3. cot if tan 5. sec if cos 1 5 1 2. cos if sec 2.5 4. csc if sin 2 4 6. tan if cot 5 3 7 Identify the quadrant or quadrants for the angle satisfying the given conditions. 7. sin 0, cos 0 8. sec 0, csc 0 9. cos 0, sin 0 10. tan 0, cot 0 11. cos 0 Give the signs of the six trigonometric functions for each angle. 12. 129 13. 406 14. 82 121 15. 16. 662 Decide whether each statement is possible or impossible. 17. sin 2 20. csc 1 0.2 18. tan 0.92 21. cos 19. sec 1 1.3 3 4 and sec 4 3 Find all the other trigonometric functions for each of the following angles. 22. tan 15 , with in Quad II 8 23. cos 3 , with in Quad III 5 24. sin 7 , with in Quad II 25 25. csc 2 , with in Quad II 26. cot 2 , with in Quad IV 27. sec 2 , with cot 0 TRIGONOMETRY PRACTICE TEST: Chapter 1 NAME: _______________ Trigonometric Functions Find the distance between each of the following pairs of points. 1. 3. A 4, 2 and B 1, 6 2. C 6,3 and D 2, 5 Using both the distance formula and the Pythagorean Theorem, determine if the triangle made up of the following points is a right triangle or not. E 2, 2 , F 8, 4 and G 2,14 Find the domain and range. Write each set using interval notation. 4. y 9x 2 5. y x 6. y x 1 7. Find the angle of smallest possible positive measure coterminal with the following angles. 8. 51 11. A pulley is rotating 320 times per minute. Through how many degrees does a point on the pulley move in 9. 174 10. 792 2 seconds? 3 Convert decimal degrees to degrees, minutes, seconds, and convert degrees, minutes, seconds to decimal degrees. Round to the nearest second or the nearest thousandth of a degree, as appropriate. 12. 472511 13. 61.5034 Find the measure of each marked angle. B 14. 15. 8x 9 x 4 12x 14 6x 4x A C Find all unknown parts (angles and side lengths) in the pair of similar triangles. 16. Z 32 T 20 14 X 41 V 11 U Y 22 Find the values of all the trigonometric functions for an angle in standard position having the following point on its terminal side. Rationalize the denominator when applicable. 17. 3,3 18. 1, 3 19. 6,0 20. 8,15 21. 0,1 22. 2 23. 2 ,2 2 Find the values of all the trigonometric functions for an angle in standard position having its terminal side defined by the equation 5 x 3 y 0, x 0 . Evaluate each expression. 24. 4 sec 180 2 sin 2 270 25. cot 2 90 4 sin 270 3 tan 180 Decide whether each statement is possible or impossible. 3 4 and csc 4 3 26. sin 28. tan 1.4 2 3 27. sec 29. cos .25 and sec 4 Find all the other trigonometric function values for each angle. Rationalize the denominators when applicable. 30. sin 3 and cos 0 5 32. sec 5 with in Quad IV 4 31. cos 5 with in Quad III 8 Solutions EF 2 34, EG 4 17, FG 2 34 1. AB 5 2. CD 4 5 3. 272 272 rt 4. 7. 11. Domain : , Range : , Domain : 5,5 Range : 3,3 1280 12. 5. 8. Domain : , Range : 0, 309 47.420 9. 13. mA 60 15. mB 80 17. 16. sec 1, csc und 613012 10. 18. 3 1 , cos 2 2 tan 3, cot sec 2, csc 20. 3 3 2 3 3 15 8 , cos 17 17 15 8 tan , cot 8 15 17 17 sec , csc 8 15 sin 72 58,58 14. mZ 32, XZ 40 sin sin 0, cos 1 tan 0, cot und 186 my 107 mU sec 2, csc 2 19. Range : 0, mV 41, ZY 28 mC 40 2 2 sin , cos 2 2 tan 1, cot 1 Domain : 1, 6. sin 1, cos 0 21. tan und , cot 0 22. sec und , csc 1 5 34 3 34 , cos 34 34 5 3 tan , cot 3 5 2 2 , cos 2 2 tan 1, cot 1 sin sec 2, csc 2 sin 23. 24. 6 25. 4 34 34 , csc 3 5 sec possible 26. 1 sin 1 csc 1 27. impossible sec 1or sec 1 reciprocals 28. possible tan sin 30. 32. 29. 3 22 , cos 5 5 tan 66 66 , cot 22 3 sec 5 22 5 3 , csc 22 3 3 4 sin , cos 5 5 3 4 tan , cot 4 3 5 5 sec , csc 4 3 impossible not reciprocals sin 31. tan 39 5 , cos 8 8 39 5 39 , cot 5 39 8 8 39 sec , csc 5 39