NOTES 2.5, 6.1 6.3 Name: 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 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. Direct variation as th n power y n kx and 0 k - th y varies directly as the n power of x - th y is directly proportional to the n power of x B is directly proportional to the square of L. If L = 15, then B = 1350. Inverse variation k y 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. 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. 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. 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? Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 2
LESSON 6.1 ANGLE MEASURE Trigonometry Angles Note: Labeled with Greek letters:,,,... Measurement of triangles 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 Coterminal Angles Angles with the same initial and terminal sides. and 360n and 2n 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 3. 540 Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 3
Radian Measure 360 2 rad 180 rad One radian is the measure of a 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, 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 is 90 rad 60 2 s where r is measured in radians. 3 rad Degrees to Radians Multiply by 180 Example: Radians to Degrees Multiply by 180 Example: Practice Problems: Convert the following angles from degrees to radians and from radians to degrees without using a calculator. 4. 150 7 6. 240 5. 6 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 11. 0.57 Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 4
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. 13 12. 6 3 13. 4 14. 2 3 Length of a Circular Arc In a circle of radius r, the length s of an arc that subtends a central angle of radians is: s r Note: must be in radians. Example: Review Problem 15: Find the following arc lengths using geometry then use s r to validate your answers. Given CB 24in, ma0b60, find the following arc lengths using 2 methods 1. Circumference = 2. Length of AB = A C O 24 in 60 B 3. Length of CA = D mab mca mcdb madb madc 4. Length of CDB = 5. Length of ADB = 6. Length of 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. Find the radius of the circle if an arc of length 8 in on the circle subtends a central angle of 4. 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? Angular Speed Linear Speed 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. The linear velocity of a point on a rotating object is the distance per unit time that the point travels along its circular path. t s v t Note: Relationship between Linear and Angular Speed 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. If a point moves along a circle of radius r with angular speed, then its linear speed v is given by: v r 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. (a) Find the linear velocity of the chain. 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) 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. opp sin csc hyp hyp opp SOH CAH TOA adj cos sec hyp hyp adj opp tan cot adj adj opp Review: Special Right Triangles x 45 x 2 30 90 45 x 3 2x x 90 60 x Practice Review Problems: Evaluate the following 1. a 4 b c b 45 c 2. a b 6 2 c 90 a 45 3. a b c 10 4. a 7 3 b c 5. a b hat c 6. a b c ipod 7 Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 8
7. a 4 b c b 30 c 8. a b 6 2 c 9. a b c 10 90 60 a 10. a 7 3 b c 11. a b hat c 12. a b c ipod 7 Practice Problems: Evaluate the six trig functions at each real number without using a calculator. 1. sin 17 csc 4 cos sec tan cot 2. 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. csc45 5. tan30 6. sec30 Practice Problems: Evaluate the given expression without using a calculator. Leave your answer in simplest radical form. 7. sin 60 cos30 8. tan 45 cot(60) 9. tan 60sec60 10. 15sin 30cos 45 Practice Problems: Solve the right triangle. B a b ma mb a c c mc C 34.2 b=19.4 A Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 10
12. a ma 100 b mb 75 c mc 13. a ma 72.3 6 b c mb mc Angles of Elevation and Depression 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 x, a point on the terminal side of and 2 2 0. r x y y y sin csc r, y 0 r y x cos sec r, x 0 r x y tan, x 0 cot x, y 0 x y Practice Problem 1: Let 4, 3 be a point on the terminal side of. Find: sin cos csc sec tan cot Practice Problem 2: Let 1 3 3, 7 be a point on the 2 4 terminal side of. Find: sin cos csc sec tan cot Signs of the Trigonometric Functions Use All Student Take Calculus to figure out in which quadrant each trig function has a positive value. Students sin/csc positive All Take tan/cot positive Calculus cos/sec positive Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 12
II ( -, + ) sin : cos : tan : 2 ( +, + ) csc : sin : csc : sec : cos : sec : cot : tan : cot : I 0 sin : cos : tan : ( -, - ) III csc : sec : cot : sin : cos : tan : 3 2 csc : sec : cot : ( +, - ) IV Practice Problem 3: Find the value of the six trigonometric 15 functions. Given: tan ; 8 sin 0 sin cos tan csc sec cot Practice Problem 4: Find the value of the six trigonometric functions. Given: cot is 3 undefined; 2 2 sin cos tan csc sec cot Practice Problem 5: Evaluate the following: a. sin 0 b. sin c. sin 2 d. 3 sin 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 ' rad ' 180 deg ree Quadrant III ' rad ' 180 deg ree Quadrant IV ' 2 rad ' 360 deg ree Practice Problem 6: Find the reference angle for the following. Graph the angle a. 309 ' b. 145 ' ' c. 7 ' d. 4 11 ' 3 Evaluating Trigonometric Functions of any Angle 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 1 sin csc 1 cos sec 1 tan cot 1 csc sin 1 sec cos 1 cot tan Quotient Identities sin tan cos cos cot sin Pythagorean Identities 2 2 sin cos 1 2 2 1 tan sec 2 2 1cot csc Practice Problem 7: Evaluate the trig functions a. 4 b. tan cos 210 3 c. 11 csc 4 d. cot 2 Practice Problem 8: Find the indicated trig function a. 5 cos and Quad III 8 sec b. 3 sin and Quad IV 5 cos Practice Problem 9: Find two exact solutions of the equation. in degrees 0 360 and radians 0 2. a. cot 1 b. tan 2.3545. Round to 2 decimal places. Mrs. Nguyen Honors Algebra II Chapter 2.5, 6.1 6.3 Notes Page 15