9.2 Angles and Radian Measure EEssentialssential QQuestionuestion How can you fi nd the measure of an angle in radians? Let the vertex of an angle be at the origin, with one side of the angle on the positive x-axis. The radian measure of the angle is a measure of the intercepted arc length on a circle of radius 1. To convert between degree and radian measure, use the fact that π radians — = 1. 180° Writing Radian Measures of Angles Work with a partner. Write the radian measure of each angle with the given degree measure. Explain your reasoning. a. y b. y ° 90 120° 60° 135° 45° radian 150° 30° measure π 0° 180° 360° x x degree ° ° measure 210 330 225° 315° 240° 300° 270° Writing Degree Measures of Angles Work with a partner. Write the degree measure of each angle with the given radian measure. Explain your reasoning. y degree 5π 4π measure 9 9 7π 2π radian 9 9 measure x 11π 16π 9 9 13π 14π REASONING 9 9 ABSTRACTLY To be profi cient in math, you need to make sense CCommunicateommunicate YourYour AnswerAnswer of quantities and their 3. How can you fi nd the measure of an angle y relationships in problem in radians? situations. 4. The fi gure shows an angle whose measure is 30 radians. What is the measure of the angle in degrees? How many times greater is 30 radians x than 30 degrees? Justify your answers. 30 radians Section 9.2 Angles and Radian Measure 469 hhsnb_alg2_pe_0902.inddsnb_alg2_pe_0902.indd 469469 22/5/15/5/15 11:48:48 PPMM 9.2 Lesson WWhathat YYouou WWillill LLearnearn Draw angles in standard position. Find coterminal angles. Core VocabularyVocabulary Use radian measure. initial side, p. 470 terminal side, p. 470 Drawing Angles in Standard Position standard position, p. 470 coterminal, p. 471 In this lesson, you will expand your study of angles to include angles with measures that can be any real numbers. radian, p. 471 sector, p. 472 central angle, p. 472 CCoreore CConceptoncept Previous Angles in Standard Position ° y radius of a circle In a coordinate plane, an angle can be formed 90 circumference of a circle terminal by fi xing one ray, called the initial side, and side rotating the other ray, called the terminal side, 0° about the vertex. 180° vertex initial x 360° An angle is in standard position when its vertex side is at the origin and its initial side lies on the positive x-axis. 270° The measure of an angle is positive when the rotation of its terminal side is counterclockwise and negative when the rotation is clockwise. The terminal side of an angle can rotate more than 360°. Drawing Angles in Standard Position Draw an angle with the given measure in standard position. a. 240° b. 500° c. −50° SOLUTION a. Because 240° is 60° b. Because 500° is 140° c. Because −50° is more than 180°, the more than 360°, the negative, the terminal terminal side is 60° terminal side makes side is 50° clockwise counterclockwise past one complete rotation from the positive the negative x-axis. 360° counterclockwise x-axis. plus 140° more. y y y ° 240 140° x x x ° −50° 60° 500 MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Draw an angle with the given measure in standard position. 1. 65° 2. 300° 3. −120° 4. −450° 470 Chapter 9 Trigonometric Ratios and Functions hhsnb_alg2_pe_0902.inddsnb_alg2_pe_0902.indd 470470 22/5/15/5/15 11:48:48 PPMM Finding Coterminal Angles In Example 1(b), the angles 500° and 140° are coterminal because their terminal sides coincide. An angle coterminal with a given angle can be found by adding or STUDY TIP subtracting multiples of 360°. If two angles differ by a multiple of 360°, then the Finding Coterminal Angles angles are coterminal. Find one positive angle and one negative angle that are coterminal with (a) −45° and (b) 395°. SOLUTION There are many such angles, depending on what multiple of 360° is added or subtracted. a. −45° + 360° = 315° b. 395° − 360° = 35° −45° − 360° = −405° 395° − 2(360°) = −325° y y −325° 35° −45° x ° x 315° 395 −405° MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Find one positive angle and one negative angle that are coterminal with the given angle. 5. 80° 6. 230° 7. 740° 8. −135° STUDY TIP Using Radian Measure Notice that 1 radian Angles can also be measured in radians. To defi ne y is approximately equal a radian, consider a circle with radius r centered at to 57.3°. the origin, as shown. One radian is the measure of r 180° = π radians an angle in standard position whose terminal side r intercepts an arc of length r. ° —180 = π 1 radian Because the circumference of a circle is 2πr, there 1 radian x are 2π radians in a full circle. So, degree measure ° ≈ 57.3 1 radian and radian measure are related by the equation 360° = 2π radians, or 180° = π radians. CCoreore CConceptoncept Converting Between Degrees and Radians Degrees to radians Radians to degrees Multiply degree measure by Multiply radian measure by π radians 180° — . — . 180° π radians Section 9.2 Angles and Radian Measure 471 hhsnb_alg2_pe_0902.inddsnb_alg2_pe_0902.indd 471471 22/5/15/5/15 11:48:48 PPMM Convert Between Degrees and Radians Convert the degree measure to radians or the radian measure to degrees. π a. 120° b. − — 12 READING The unit “radians” is often SOLUTION omitted. For instance, the π radians π π 180° π a. 120° = 120 degrees — b. − — = − — radians — measure − — radians may ( 180 degrees) 12 ( 12 )( π radians) 12 π be written simply as − — . 2π 12 = — = −15° 3 CConceptoncept SSummaryummary Degree and Radian Measures of Special Angles The diagram shows equivalent degree and y π radian π π radian measures for special angles from 2 2 measure 0° to 360° (0 radians to 2π radians). 3π 3 3 π 4 90° 4 5π 120° 60° π You may fi nd it helpful to memorize the 6 135° 45° 6 ° ° equivalent degree and radian measures of 150 degree 30 special angles in the fi rst quadrant and for π measure 0° 0 x π 180° 90° = — radians. All other special angles 360° 2π 2 shown are multiples of these angles. 210° 330° 7π 225° 315° 11π 6 240° 300° 6 5π 270° 7π 4 π π 4 4 π 5 3 3 3 2 MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Convert the degree measure to radians or the radian measure to degrees. 5π 9. 135° 10. −40° 11. — 12. −6.28 4 A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle θ of a sector is the angle formed by the two radii. There are simple formulas for the arc length and area of a sector when the central angle is measured in radians. CCoreore CConceptoncept Arc Length and Area of a Sector The arc length s and area A of a sector with sector radius r and central angle θ (measured in radians) are as follows. r arc Arc length: s = rθ length s 1 central Area: A = — r 2θ 2 angle θ 472 Chapter 9 Trigonometric Ratios and Functions hhsnb_alg2_pe_0902.inddsnb_alg2_pe_0902.indd 472472 22/5/15/5/15 11:49:49 PPMM Modeling with Mathematics A softball fi eld forms a sector with the dimensions shown. Find the length of the outfi eld fence and the area of the fi eld. SOLUTION outfield 1. Understand the Problem You are given the fence dimensions of a softball fi eld. You are asked to fi nd the length of the outfi eld fence and the area of the fi eld. 200 ft 2. Make a Plan Find the measure of the central angle in radians. Then use the arc length and area of a sector formulas. 90° 3. Solve the Problem 200 ft Step 1 Convert the measure of the central angle to radians. π radians COMMON ERROR 90° = 90 degrees — ( 180 degrees) You must write the π measure of an angle = — radians in radians when using 2 these formulas for the arc length and area of Step 2 Find the arc length and the area of the sector. a sector. 1 Arc length: s = rθ Area: A = — r 2θ 2 π 1 π = 200 — = — ( 200)2 — ( 2 ) 2 ( 2 ) = 100π = 10,000π ANOTHER WAY ≈ 314 ≈ 31,416 Because the central angle is 90°, the sector The length of the outfi eld fence is about 314 feet. The area of the fi eld 1 — is about 31,416 square feet. represents 4 of a circle with a radius of 200 feet. So, 4. Look Back To check the area of the fi eld, consider the square formed using the two 1 1 s = — πr = — π 4 ⋅ 2 4 ⋅ 2 (200) 200-foot sides. = 100π By drawing the diagonal, you can see that and the area of the fi eld is less than the area of the 200 ft 1 1 square but greater than one-half of the area of A = — ⋅ πr 2 = — ⋅ π (200)2 4 4 the square. = π 10,000 . 1 — 2 ⋅ (area of square) area of square 90° 200 ft 1 ? ? — (200)2 < 31,416 < 2002 2 20,000 < 31,416 < 40,000 ✓ MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 13. WHAT IF? In Example 4, the outfi eld fence is 220 feet from home plate.