An angle on the coordinate plane is in standard form when the vertex is on the origin and one ray lies on the positive x-axis.
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1 Name: Topic: Main Ideas/Questions Notes/Eamples Date: Class: Angles in Standard Form y θ An angle on the coordinate plane is in standard form when the verte is on the origin and one ray lies on the positive -ais. The ray on the -ais is called the. The other ray is called the. Counterclockwise rotations result in angle measures. Clockwise rotations result in angle measures. DEGREES The most common unit of measure for angles is the, ( ). A degree is equivalent to of a a full rotation. 90 Sketch an angle in standard form with the given measure y y y / y y y RADIANS r θ θ = 1 radian r A radian is a unit of angle measure based on arc length. One radian is defined as the measure of the angle formed when the radius is equivalent to the length of the intercepted arc. Recall that the circumference of a circle is 2πr, therefore: 360 = ; 180 = Converting Degrees Radians Converting Radians Degrees radians Radians = Degrees 180 Degrees = Radians 180 radians Gina W ilson (All Things Algebra, LLC), 2018
2 Converting DEGREES to RADIANS Convert each angle measure to radians Converting RADIANS to DEGREES Convert each angle measure to degrees COTERMINAL ANGLES Angles in standard position with the same terminal side are coterminal angles. Determine whether the two angles are coterminal and and -170 y and and Give two coterminal angles for each given angle, one positive and one negative Gina W ilson (All Things Algebra, LLC), 2018
3 Name: Topic: Date: Class: Main Ideas/Questions DEGREE- MINUTE-SECOND FORM (DMS) Notes/Eamples Some angles have decimal degrees (for eample, 26.5 ). Angles with decimals can be epressed using degree-minute-second form (DMS) in which the degrees are subdivided into minutes and seconds as follows: one degree = minutes; one minute = seconds *one degree = seconds Symbols for degree, minutes, and seconds: DEGREE MINUTE SECOND Follow the eample below to convert to degree-minute-second form. Converting DECIMAL DEGREES TO DMS = = Therefore, = Write each angle measure in DMS form
4 Follow the eample below to convert to decimal degrees. Converting DMS TO DECIMAL DEGREES / = / = Therefore, = = Write each angle measure in decimal degree form
5 Name: _ Date: Per: Unit 5: Trigonometric Functions Homework 1: Angle Measures ** This is a 2-page document! ** Directions: Convert each angle measure to radians Directions: Convert each angle measure to degrees. 7. 3p p p p p p 4 Directions: Give two coterminal angles for each given angle, one positive and one negative p p p 6
6 Directions: Write each angle measure in Degree-Minute-Second (DMS) form Directions: Write each angle measure in decimal degree form
7 Name: Date: Topic: Main Ideas/Questions ARC LENGTH Notes/Eamples If θ is a central angle in a circle with radius r, then the length of the intercepted arc, s, is given by: (where θ is measured in radians) Class: r θ s Eamples Find the length of the intercepted arc given the angle measure and radius. Give your answer in terms of π and rounded to the nearest tenth ; r = 3 ft 2. ; r = 14.5 in ; 4. 2 ; r = 6.2 yd r = 0.5 yd Convert to radians! ; r = 8.4 m ; r = 10.5 mm ; r = 1.8 ft ; r = 25 km 9. The central angle θ in a circle of radius 6 meters has an intercepted arc length of 10 meters. Find the measure of θ in radians and in degrees.
8 10. A merry go round rotates 2808 per ride. How far would a rider seated 8 feet from the center of the merry go round travel during the ride? 11. Cincinnati, Ohio is directly north of Atlanta, Georgia. Cincinnati has a latitude of 39.1 N and Atlanta has a latitude of 33.7 N. If the earth has a radius of 3,960 miles, how far apart are these cities? AREA OF SECTORS The area A of a sector of a circle with radius r and central angle θ is given by: r θ A (where θ is measured in radians) Find the area of each sector. Eamples cm ft km 23.6 m 16. The area of a sector of a circle with a central angle of 240 is 134 ft 2. Find the radius of the circle. 17. The windshield wiper blade to the left is 32 inches long. If the wiper sweeps through an angle of 125, find the area swept by the blade. 21 in
9 Name: _ Date: Per: Unit 5: Trigonometric Functions Homework 2: Arc Lengths & Area of Sectors ** This is a 2-page document! ** Directions: Find the length of each intercepted arc given the angle measure and radius. Give your answer in terms of π and rounded to the nearest tenth ; 8 r = 10 mm 2. 7 ; r = 17.5 yd 5 3. ; r = 2 mi ; r = 18 ft ; r = 0.9 cm ; r = 4.5 yd ; r = 6.6 km ; r = 2.2 m 9. An intercepted arc has a length of 19 yards. If the radius is 4 yards, find the measure of the central angle in radians and degrees. 10. The end of minute hand on a clock is 5.5 inches from the center. How far will the minute hand travel over three full days?
10 Directions: Find the area of each sector ft mi km 7.1 cm yd 21.4 m 17. The sector of a circle has a central angle of 144 and an area of 70.7 square meters. Find the radius of the circle. 18. A circle with a radius of 15 yards contains a sector with an area of 609 yd 2. Find the measure of the central angle of the sector in both radians and degrees. 19. A wall clock is equally divided into 12 sections. If the clock reads 8:00 and has a diameter of 12.5 inches, find the area of the smaller sector formed by the minute and hour hands.
11 Name: Date: Topic: Main Ideas/Questions CIRCULAR MOTION s r θ Notes/Eamples Arc length can be used to analyze circular motion. Suppose an object moves along a circular path s with radius r: The rate at which the object moves along the path is called its linear speed, v. Linear Speed Formula: Class: The rate at which the central angle changing along the path is called its angular speed, ω. Angular Speed Formula: Eamples Use for questions 1-2: A bicycle tire with a radius of 14 inches rotates at a rate of 125 revolutions per minute (rpm). 1. Find the linear speed of the tire in 2. Find the angular speed of the inches per minute. tire in radians per minute. Recall: Each rotation (or revolution) has a circumference of 2πr and a central angle of 2π radians. Use for questions 3-4: A CD with a diameter of 120 millimeters rotates a rate of 45 revolutions per minute. 3. Find the linear speed of the CD 4. Find the angular speed of the in millimeters per minute. CD in radians per minute. 5. A 16-inch diameter tire on a car is making 500 revolutions per minute. Find the linear speed of the tire miles per hour.
12 6. A circular saw blade with a diameter of 9 inches rotates at 2800 revolutions per minute. Find the angular speed of the blade in radians per second. 7. A windmill has blades that are 14 feet long. If the windmill is rotating at 5 revolutions per second, find the linear speed of the tips of the blades in miles per hour. Using ANGULAR SPEED to find Linear SPEED The linear speed, v, can also be found as follows: v = = = = Therefore, you can use the angular speed, ω, to find the linear speed, v. 8. A ceiling fan with 25-inch blades rotates at 40 rpm. Find the linear speed of the tips of the blades in feet per second. 9. Ryan is riding a bicycle whose wheels are 28 inches in diameter. If the wheels rotate at 130 rpm, find the linear speed in miles per hour in which he is traveling.
13 Name: _ Date: Per: Unit 5: Trigonometric Functions Homework 3: Linear & Angular Speeds ** This is a 2-page document! ** Use for questions 1 and 2: An SUV tire with a radius of 12.5 inches rotates at a rate of 545 revolutions per minute. 1. Find the linear speed of the tire in inches per 2. Find the angular speed of the tire in radians minute. per minute. Use for questions 3 and 4: The fan on a pool heater has a diameter of 107 centimeters and rotates at a rate of 212 revolutions per minute. 3. Find the linear speed of the fan in 4. Find the angular speed of the fan in radians centimeters per minute. per minute. 5. A 56-mm wheel on a skateboard makes 430 revolutions per minute. Find the linear speed of the wheel in meters per second. 6. A 12-inch bicycle tire is making 175 revolutions per minute. Find the linear speed in miles per hour.
14 7. A record with a diameter of 12 inches rotates at 33 revolutions per minute. Find the angular speed of the record in radians per second. 8. An oscillating fan has blades that measure 8 inches long. If the fan rotates at a speed of 9 revolutions per second, find the linear speed of the ends of the blades in miles per hour. 9. Darlene is driving a car whose wheels have an 11 inch radius. If Darlene s wheels are rotating at a rate of 378 rpm, find the linear speed in miles per hour in which she is traveling. 10. A DVD with a radius of 6 centimeters rotates at a speed of 243 revolutions per minute. Find the linear speed in centimeters per second. 11. A lawnmower blade has a diameter of 30 inches. If the blade rotates at a speed of 126 revolutions per minute, find the linear speed of the tips of the blades in feet per second.
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