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Review of Right Triangle Trigonometry Name: UNIT 7:Trigonometry I Do Now: 1. For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is 12 and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9. Notes: Sin = Cos = Tan = 1. Use the diagram below to answer the evaluate a-f. F a) Sin D b) Cos D c) Tan D 5 4 d) Sin F e) Cos F f) Tan F D 3 E 2. In PQR , mP 90 , PR 5 and QR 13 . Find 3. In CAT , mA 90 , AT 8 and CT 10 . Find sinC. cosR. 4. Find x to the nearest tenth. 5. Find x to the nearest tenth. 12 40 x x 32 50 pg. 0 6. Find x to the nearest degree: 3 7. A 10-foot tree is supported in a vertical position by a rope running from the top of the tree to a stake in the ground. If the angle that the rope makes with the ground is 63o, find the length of the supporting rope to the nearest foot. x 4 In # 8 – 15, determine the quadrant in which an angle of the given measure lies. 8. 140o 9. -430o 10. 210o 11. 97o 12. 315o 13. -168o 14. -200o 15. 600o In #16– 23, draw an angle of the given measure, indicating the direction of the rotation by the arrow. Find the angle coterminal with an angle of the given measure. 16. 100o 17. -50o 18. 210o 19. -300o 20. 470o 21. -800o 22. -200o 23. 60o 24. Which pair of angles are coterminal? (1) 30o and -30o (2) 150o and -210o (3) 200o and 160o (4) -100o and -80o pg. 1 Homework on Review of Right Triangle Trigonometry 1. Which ratio represents tan A in the diagram below? (1) (3) (2) (4) 2. In the diagram below of right triangle KTW, , , and What is the measure of (1) 33o . , to the nearest degree? (2) 34o (3) 35o (4) 36o 3. Which pair of angles are coterminal? (1) 40o, 140o (2) -30o, -150o (3) 310o, -50o (4) 120o, 240o In # 4 – 7, determine the quadrant in which the angle terminates. 4. -85o 5. 468o 6. -1000o 7. 219o 8. A ladder 10 feet in length leans against a tree, making an angle of 28o with the ground. How far from the tree was the ladder placed, to the nearest hundredth? 9. Put the equation of the circle in center-radius form. Determine the center and radius: 10. If f ( x) x 2 2 x 11 & g ( x) 3x 2 , find f ( g ( x)) . x 2 y 2 6 x 16 y 2 0 pg. 2 The Unit Circle, Signs of the Quadrants, and Quadrantal Angles Do Now: The conjugate of is (1) (2) (3) (4) Notes: Signs of the Quadrants Unit Circle: Center = Radius = P (x, y) = In #1-4, name the quadrants in which an angle of measure could lie when: 1. cos 0 2. sin 0 3. tan 0 4. sin 0 In #5 – 10, name the quadrant in which an angle of measure could lie when: 5. sin 0 and cos 0 6. sin 0 and cos 0 8. cos 0 and sin 0 9. cos 3 10 and tan 0 5 7. tan 5 and cos 0 6 10. sin 0 and tan 0 pg. 3 11. If P is a point that lies on the unit circle and P(.6, .8) , then find: 12. In the figure, a unit circle is drawn and mCOA . Name the line segment whose length is: a) sin y a) sin B A b) cos O P C x c) tan b) cos c) tan Complete the following table, giving the functional value of each quadrantal angle. If a function is not defined for an angle measure, write “undefined.” A Sin A 0o 90 o 180 o 270 o 360 o Cos A Tan A In #13 - 18, find the exact numerical value of the expressions given below. 13. cos 270 16. sin 90 o cos 0 o tan 180 o sin 90 o 14. (cos 0 o ) 2 tan 180o 15. 17. cos180 o sin 270 o 18. sin 720 cos 180 pg. 4 HW on Unit Circle, Signs of Quadrants, Quadrantal Angles 1. If is negative and (1) I 2. If is negative, in which quadrant does the terminal side of (2) II (3) III and (1) I 3. If 4. If (1) I 5. If (1) I and II 6. If (1) I and II (4) IV lies in which quadrants? (2) II and III and terminate? (3) III , then angle (1) I and II (4) IV , in which quadrant does (2) II lie? (3) I and III (4) III and IV , in which quadrant does angle x lie? (2) II (3) III , in which quadrants may angle (2) II and III , then angle (4) IV terminate? (3) I and III (4) III and IV may terminate in Quadrant (2) II and III (3) I and III (4) III and IV 1 3 7. If P , lies on the terminal side of of a unit circle, find: 2 2 a) the quadrant terminates in b) the value of sin c) the value of cos d) the value of tan pg. 5 8. An angle whose measure is -214o is coterminal with all of the following except (1) -574o (2) 34o (3) 146o (4) 506o 5 12 9. If is an angle in standard position and P , is a point on the unit circle on the terminal side of , 13 13 what is the value of tan ? (1) 12 5 (2) 5 13 (3) 5 12 (4) 12 13 10. Find the solution set to the inequality and graph it on the number line below: 10 2 x 4 4 11. Solve for x and express your answer in simplest radical form: 4 3 7 x x 1 pg. 6 Special Angles of 30o, 60o, and 45o Do Now: 1. If cos 3 2 , in which quadrants can terminate? 6 2. Simplify: 3 2 5 Notes: 60 2 45 2 1 1 45 30 1 3 Fill in the table below: Sin 30o 45o 60o Cos Tan Find the exact numerical value of the expression. 1. sin 30 o cos 60 o 2. (sin 60 o )(cos 60 o ) 3. (tan 60 o ) 2 4. sin 45 o cos 45 o pg. 7 5. cos 60 0 (cos 0 0 cos180 0 ) 9. (cos 45o ) 2 6. 2 sin 30 cos 30 10. (tan 60 o )(tan 30 o ) 13. If is the measure of an acute angle and sin (1) cos 3 2 (2) tan 1 7. (tan 2 45 )(sin 45 ) 8. tan 45o 2 cos 60 o 11. (cos 60 o 2 tan 45o ) 2 12. (tan 60 0 ) 2 sin 180 0 3 , then: 2 (3) cos 1 2 (4) tan 3 3 14. If 30 o , which expression has the largest numerical value? (1) sin (2) cos (3) tan (4) (cos )(tan ) (3) 45 o (4) 60 o 15. If sin 3 cos , then can equal: (1) 0 o (2) 30 o pg. 8 Special Angles Homework In #1 – 6, find the exact value of each expression: 1. tan 30º + cos 60º 2. (tan 45º)(cos 45º) 3. cos 60º + sin 30º - tan 45º 4. sin2 45º • cos 45º 5. (cos 60º)2 +(sin 30º)2 6. (sin 0 0 )(tan 30 0 ) cos 0 0 7. If sin 0 and tan 0 , what might be the measure of ? (1) 81o (2) 128o (3) 207o (4) 313o 8. Which of the following is a false statement? (1) (2) (3) (4) Tan is undefined whenever cos equals zero. Tan equals zero whenever sin equals zero. Sin can equal cos in Quadrant I or Quadrant III of the unit circle. Sin can equal cos only in Quadrant I of the unit circle. 9. If sin 3 cos , then can equal (1) 0 0 (2) 30 0 (3) 45 0 (4) 60 0 10. The value of 2(sin 30 0 )(cos 30 0 ) is equal to the value of (1) sin 60 0 (2) cos 60 0 1 a 11. Simplify: 1 1 2 a (3) sin 90 0 1 12. Solve for x: (4) tan 30 0 x2 x pg. 9 Reference Angles Do Now: 1. Which angle is coterminal to -400o? (1) 40o (2) -40o (3) 140o (4) -220o Notes: What is a reference angle?? In #1-8 write the reference angle for each given angle. 1. 100o 2. 340o 3. 248o 4. 98o 5. 200o 6. 45 7. 745o 8. 460 In 9 – 16, express the given function as a function of a positive acute angle. 9. sin100 10. cos150 11. tan 300 12. cos190 13. tan145 14. sin 650 15. cos(200 ) 16. tan(80 ) 19. tan(120 ) 20. cos 570 In 17 – 20, find the exact function value. 17. cos 315 18. sin(135 ) pg. 10 HW on Reference Angles In # 1 – 4, find the EXACT value: 1. sin120 3. tan 225 2. cos 315 4. cos 390 sin 270 2 In # 5 – 8, express each function as a function with a positive acute angle. 6. tan 50 5. sin 320 7. cos 560 8. sin 400 1 3 9. If P , lies on the terminal side of on a unit circle, what can be? 2 2 (1) 60o (2) 120o (3) 210o (4) 240o (3) 228o (4) -228o 10. What is the reference angle for -132o? (1) -48o (2) 48o 11. If f ( x) sin 2 x cos x , evaluate: a) f 180 b) f 30 12. Is r directly or inversely proportional to s? 13. If z1 2 5i and z2 5 2i , find and graph z1 z2 . Evaluate z1 z2 Find x. r s 4 6 12 2 x 3 pg. 11 Converting to and from Radian Measure!! Do Now: 1. For the angle -500o, find: a) the reference angle 2. Given the diagram below, circle O is a unit circle. Find the line segment that represents: b) an angle co-terminal to it y a) sin B A b) cos O P C x c) tan CONVERTING: In 1-8, convert each degree measure radian measure in terms of . 1. 90 2. 45 3. 60 4. 30 5. 120 6. 150 7. 75 8. 160 In 9 – 16, find the degree measure of an angle of the given radian measure. 9. 3 13. 10. 5 6 14. 9 11. 10 15. 2.35 radians 12. 2 16. 3.14 radians pg. 12 HW on Converting to and from Radian Measure!! In #1 – 5, convert the following angles to radian measure. 1. 270 2. 180 3. 240 4. 620 5. 50 In #6 – 10, convert the following angles to degree measure. 6. 2 7. 7 4 8. 9. 5 3 10. 2 11. Convert 5.21 radians to degrees. Round your answer to the nearest thousandth. 12. What is the domain of y (1) 3 x 3 1 9 x2 ? (2) 3 x 3 (3) x 3 or x 3 (4) x 3 or x 3 13. If one of the roots to the equation x 2 kx 6 0 is 3, find k. (1) 1 (2) -1 (3) -2 (4) 5 14. If f ( x) x 2 and g ( x) x 2 3 , find the value of f g (1) . 15. What type of roots will the equation x 2 4 x 8 have? (1) real, rational, and equal (2) real, rational, and unequal (3) real and irrational (4) imaginary 16. Solve for x by completing the square. Place your answer in simplest radical form. 2 x 2 12 x 6 0 pg. 13 Evaluating Trigonometric Functions!! Do Now 1. If P(0.6, 0.8) lies on the terminal side of on the unit circle, find: c) cos b) tan a) sin 2. Simplify: 2 3 16a7bc24 3. Simplify and express with positive exponents: 25a 6bc 1 15ab8c 3 Evaluating in Radians and Degrees! In 1-8, find the exact value of the trigonometric function. s 1. cos 2. tan 45 3 5. sin 225 6. tan 5 6 3. sin 2 3 7. cos 225 4. cos 4 3 8. tan 2 9. If a function f is defined as f ( x) cos 3x find the numerical value of: a) f 2 b) f ( ) c) f (45 ) d) f (30 ) In 10 – 12, find the numerical value of f ( ) for the given function f. x 10. f ( x) tan 3 11. f ( x) sin x cos 2 x 12. f ( x) sin x cos 2 x pg. 14 HW on Evaluating Trigonometric Functions in Radians and Degrees! In #1 – 4, find the exact value of each. 2. cos 30 1. tan3 In 5-7, find the numerical value of f 60 3. sin 5 3 for the given function f. 6. f ( x) sin 12 x cos5x 5. f ( x) sin 2 x 4. sin 510 7. f ( x) tan x tan 2 x 8. An angle whose measure is 3 has the same terminal side as an angle whose measure is (1) 2 3 (2) 5 3 (3) (4) 3 5 3 9. Which expression is undefined? (1) sin 2 (2) cos 2 (3) tan 2 (4) tan 4 10. If sin x > 0 and cos x < 0, x must be the measure of an angle in Quadrant (1) I or II (2) II or IV (3) I or IV (4) II or III In #11-13, write each expression as a function of a positive acute angle (reference angle). 11. sin138 o 12. cos490 o 3 13. tan 4 14. Brianna lets out 300 feet of string while she is flying her kite. Assuming that the string is stretched taut and makes an angle of 48 with the ground, find to the nearest foot the height of the kite above the ground. pg. 15