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Name:______________________________ NOTES 2.5, 6.1 – 6.3 Date:________________Period:_________ Mrs. Nguyen’s Initial:_________________ LESSON 2.5 – MODELING VARIATION Direct Variation y mx b when b 0 or y mx or y kx y kx - Direct variation as n th power Inverse variation and k 0 y varies directly as x y is directly proportional to x k is the constant of variation k is the constant of proportionality Express the statement as an equation. Use the given information to find the constant of proportionality. B is directly proportional to L. If L = 15, then B = 1350. y kx n and k 0 - y varies directly as the nth power of x - y is directly proportional to the nth power of x B is directly proportional to the square of L. If L = 15, then B = 1350. k or k xy x As x goes up, y goes down or As y goes up, x goes down n varies inversely as f. If n = 3, then f = 5. y Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 1 Joint variation z kxy z varies directly as x and y M varies jointly as h and n. If h = 7 and n = 9, then M = 504. Practice Problems: 1. The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. a. Write an equation that expresses this variation. b. Find the constant of proportionality if 100L of gas exerts a pressure 33.2 kPa at a temperature of 400 K. c. If the temperature is increased to 500K and the volume is decreased to 80L, what is the pressure of the gas? 2. The power P (measured in horse power, hp) needed to propel a boat is directly proportional to the cube of its speed s. An 80-hp engine is needed to propel a certain boat at 10 knots. Find the power needed to drive the boat at 15 knots. Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 2 LESSON 6.1 – ANGLE MEASURE Trigonometry Measurement of triangles Angles An angle is in standard position if… 1. its initial side is along the positive x-axis 2. its vertex is at the origin, and Note: Labeled with Greek letters: , , ,... Coterminal Angles Angles with the same initial and terminal sides. and 360n and 2 n Review 1. Central angle 2. Acute angle 3. Right angle 4. Obtuse angle 5. Arc measure 6. Arc length 7. Complementary angles 8. Supplementary angles Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co-terminal with the given angle. Sketch the angles. 1. 210 2. 45 Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 3 3. 540 Radian Measure One radian is the measure of a central Note: In a full revolution, the arc length s is equal to C 2 r s . Also, there are just over six radius lengths in a full circle. Therefore, the central angle that intercepts an arc s equal in length to the radius of a circle. C = 2r 6.28r The radian measure of an angle of one full revolution is 2 . Since one full circle has 360 , 360 2 180 rad Degrees to Radians Radians to Degrees Multiply by Multiply by rad 90 2 angle is s r where is measured in radians. 60 rad 3 rad Example: 180 180 Example: Practice Problems: Convert the following angles from degrees to radians and from radians to degrees without using a calculator. 4. 150 5. 7 6 6. 240 7. 11 30 Practice Problems: Convert the following angles from degrees to radians and from radians to degrees using a calculator and round to 3 decimal places. 8. 87.4 9. 2 10. 0.54 Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 4 11. 0.57 Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co-terminal with the given angle. Sketch the angles. 12. 13 6 13. 3 4 14. In a circle of radius r, the length s of an arc that subtends a central angle of radians is: s r Length of a Circular Arc 2 3 Example: Note: must be in radians. Review Problem 15: Find the following arc lengths using geometry then use s r to validate your answers. CB 24in , mA0 B 60 , find the Given 1. Circumference = following arc lengths using 2 methods 2. Length of AB 3. Length of = CA 4. Length of CDB = 5. Length of ADB = 6. Length of ADC = = A 60 O C B 24 in D m AB mCA mCDB m ADB m ADC Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 5 Practice Problems: Find the unknown value. 16. A central angle in a circle of radius 24 cm is subtended by an arc of length 6 cm. Find the measure of in radians. 17. 18. A bicycle’s wheels are 14 inches in diameter. How far (in miles) will the bike travel if its wheels revolve 500 times without slipping? 19. How many revolutions will a Ferris wheel of diameter 60 feet make as the Ferris wheel travels a distance of a ½ mile? 20. An ant is sitting 5 cm from the center of a c.d.. If the c.d. turns 40 , how far has the ant moved in meters? 21. A bug is on a car’s windshield wiper and is 10 inches from the base of the windshield wiper. If the bug moves 34 inches, at what angle did the windshield wiper turn? Find the radius of the circle if an arc of length 8 in on the circle subtends a central angle of 4 . The angular velocity of a point on a rotating object is the number of degrees (radians, revolutions, etc.) per unit time through which the point turns. Linear Speed The linear velocity of a point on a rotating object is the distance per unit time that the point travels along its circular path. v Note: The linear velocity depends on how far the object is from the axis of rotation, whereas the angular velocity is the same no matter where the object lies on the rotating object. Angular Speed t s t Relationship If a point moves along a circle of radius r with angular speed , then its linear speed v between Linear and is given by: v r Angular Speed Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 6 Practice Problems: Solve the following problems. 22. A woman is riding a bike whose wheels are 26 inches in diameter. If the wheels rotate at 125 revolutions per minute (rpm), find the speed at which she is traveling, in miles per hour. 23. The rear wheels of a tractor are 4 feet in diameter, and turn at 20 rpm. (a) How fast is the tractor going (feet per second)? (b) The front wheels have a diameter of only 1.8 feet. What is the linear velocity of a point on their tire treads? (c) What is the angular velocity of the front wheels in rpm? 24. The pedals on a bike turn the front sprocket at 8 radians per second. The sprocket has a diameter of 20 cm. The back sprocket, connected to the wheel, has a diameter of 6 cm. 25. Dan and Ella are riding on a Ferris wheel. Dan observes that it takes 20 seconds to make a complete revolution. Their seat is 25 feet from the axle of the wheel. (a) Find the linear velocity of the chain. (a) What is their angular velocity? (b) Find the angular velocity of the back sprocket. (b) What is their linear velocity? Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 7 LESSON 6.2 – TRIGONOMETRY OF RIGHT TRIANGLES The Trigonometric Ratios Let has an acute angle of a right triangle. The six trigonometric functions of the angle are defined below. sin opp hyp csc hyp opp cos adj hyp sec hyp adj tan opp adj cot adj opp Review: Special Right Triangles 45 30 x 2 x SOH CAH TOA 2x x 3 45 90 x 90 60 x Practice Review Problems: Evaluate the following 1. a4 b c 2. a b6 2 c 3. a b c 10 4. a7 3 b c 5. a b hat c 6. a b c iPod 7 45 c b 45 90 a Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 8 7. a4 b c 8. a b6 2 c 9. a b c 10 10. a7 3 b c 11. a b hat c 12. a b c iPod 7 30 b c 90 60 a Practice Problems: Evaluate the six trig functions at each real number without using a calculator. 1. 2. sin csc 17 4 cos sec tan cot sin csc cos sec 6 tan cot Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 9 Practice Problems: Evaluate without using a calculator. Draw and label triangles. 3. tan 60 4. csc 45 5. tan 30 6. sec30 Practice Problems: Evaluate the given expression without using a calculator. Leave your answer in simplest radical form. 7. sin 60 cos 30 8. tan 45 cot(60) 9. tan 60sec 60 10. 15sin 30 cos 45 Practice Problems: Solve the right triangle. B a c a mA b mB c mC 34.2 C b=19.4 A Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 10 12. 100 75 13. 72.3 6 Angles of Elevation and Depression a mA b mB c mC a mA b mB c mC The angle of elevation is the angle from a horizontal line UP to an object. The angle of depression is the angle from a horizontal line DOWN to an object. Practice Problems: Set up an equation for each word problem and solve. 14. Suppose you have been assigned the job of measuring the height of the local water tower. Climbing makes you dizzy so you decide to do the whole job at ground level. From a point 47.3 meters from the base, you find that you must look up at an angle of 53 degrees to see the top of the tower. How high is the tower? 15. When landing, a jet will average a 3 angle of descent. What is the altitude, to the nearest foot, of a jet on final descent as it passes over an airport radar 6 miles from the start of the runway? 16. At a point 300 feet from the base of a building, the angle of elevation to bottom of a smokestack is 40 , and the angle of elevation to the top is 55 . Find the height of the smokestack alone. 17. The distance between a plane and a building on the ground is 350 feet. The angle of depression from the plane to the building is 20 . Find the horizontal distance from the plane to the building. Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 11 LESSON 6.3 – TRIGONOMETRIC FUNCTIONS OF ANGLES Definitions of Trigonometric Functions of any Angle Let be an angle in standard position with a point on the terminal side of and r x2 y 2 0 . sin y r csc cos x r r sec , x 0 x tan y , x0 x cot Practice Problem 1: Let 4, 3 be a point on the terminal side of . Find: Practice Problem 2: Let 3 1 3 , 7 be a point on the 4 2 terminal side of . Find: Signs of the Trigonometric Functions x, y r , y0 y x , y0 y sin csc cos sec tan cot sin csc cos sec tan cot Use “All Student Take Calculus” to figure out in which quadrant each trig function has a positive value. Students All sin/csc positive Take tan/cot positive Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 12 Calculus cos/sec positive II (-,+ ) sin : cos : tan : csc : sec : sin : cos : tan : csc : sec : III (-, Practice Problem 3: Find the value of the six trigonometric 15 functions. Given: tan ; 8 sin 0 Practice Problem 4: Find the value of the six trigonometric functions. Given: cot is 3 undefined; 2 2 cot : I 2 (+ , sin : cos : tan : +) csc : sec : cot : 0 cot : sin : cos : tan : cot : , -) IV 3 ) csc : sec : (+ 2 sin csc cos sec tan cot sin csc cos sec tan cot Practice Problem 5: Evaluate the following: a. sin 0 c. sin 2 b. sin d. sin 3 2 Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 13 Definition of Reference Angle Let be an angle in standard position. Its reference angle is the acute angle ' formed by the terminal side of and the x-axis. Quadrant II Quadrant III Quadrant IV ' rad ' rad ' 2 rad ' 180 deg ree ' 180 deg ree ' 360 deg ree Practice Problem 6: Find the reference angle ' for the following. Graph the angle a. 309 ' c. 7 ' 4 Evaluating Trigonometric Functions of any Angle b. 145 ' d. 11 ' 3 To find the value of a trig function of any . 1. Determine the function value for the associated ' . 2. Depending on the quadrant in which lies, affix the appropriate sign to the function value. Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 14 FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sin 1 csc cos 1 sec tan 1 cot csc 1 sin sec 1 cos cot 1 tan Quotient Identities tan sin cos Pythagorean Identities sin 2 cos 2 1 cot cos sin 1 tan 2 sec 2 1 cot 2 csc 2 Practice Problem 7: Evaluate the trig functions a. cos 4 3 c. csc 11 4 Practice Problem 8: Find the indicated a. trig function cos 5 8 b. tan 210 d. cot sin 3 5 and Quad IV cos Practice Problem 9: Find two exact solutions of the equation. 0 2 . cot 1 2 b. and Quad III sec a. b. places. in degrees 0 360 and radians tan 2.3545 . Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 15 Round to 2 decimal