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9.2 Define General Angles and Use Radian Measure What are angles in standard position? What is radian measure? Angles in Standard Position In a coordinate plane, an angle 90° can be formed by fixing one ray called the initial side and rotating the other ray called the 180° vertex terminal side, about the vertex. An angle is in standard position if its vertex is at the origin and 270° its initial side lies on the positive x-axis. The measure of an angle is positive if the rotation of its terminal side is counterclockwise and negative if the rotation is clockwise. The terminal side of an angle can make more than one complete rotation. 0° Draw an angle with the given measure in standard position. a. 240º SOLUTION a. Because 240º is 60º more than 180º, the terminal side is 60º counterclockwise past the negative x-axis. Draw an angle with the given measure in standard position. b. 500º SOLUTION b. Because 500º is 140º more than 360º, the terminal side makes one whole revolution counterclockwise plus 140º more. Draw an angle with the given measure in standard position. c. –50º SOLUTION c. Because –50º is negative, the terminal side is 50º clockwise from the positive x-axis. Coterminal Angles Coterminal angles are angles whose terminal sides coincide. An angle coterminal with a given angle can be found by adding or subtracting multiples of 360° The angles 500° and 140° are coterminal because their terminal sides coincide. Find one positive angle and one negative angle that are coterminal with (a) –45º SOLUTION There are many such angles, depending on what multiple of 360º is added or subtracted. a. –45º + 360º = 315º –45º – 360º = – 405º Find one positive angle and one negative angle that are coterminal with (b) 395º. b. 395º – 360º = 35º 395º – 2(360º) = –325º Draw an angle with the given measure in standard position. Then find one positive coterminal angle and one negative coterminal angle. 1. 65° 65º + 360º = 425º 65º – 360º = –295º 2. 230° 230º + 360º = 590º 230º – 360º = –130º 3. 300° 300º + 360º = 660º 300º – 360º 4. = –60º 740° 740º – 2(360º) 740º – 3(360º) = 20º = –340º Radian Measure One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. Because the circumference of a circle is 2, there are 2 radians in a full circle. Degree measure and radian measure are related by the equation 360° = 2 radians or 180°= radians. Converting Between Degrees and Radians Degrees to radians Radians to degrees Multiply degree measure Multiply radian 180° measure by by 180° Degree and Radian Measures of Special Angles The diagram shows equivalent degree and radian measures for special angles for 0° to 360° ( 0 radians to 2 radians). It will be helpful to memorize the equivalent degree and radian measures of special angles in the 1st quadrant and for 90° = radians. 2 All other special angles are multiples of these angles. Convert (a) 125º to radians and (b) – π radians to 12 degrees. ( π radians ) a. 125º = 125º 180º = 25π 36 radians ( )( ) π 180º π – radians b. – = 12 π radians 12 = –15º Convert the degree measure to radians or the radian measure to degrees. 5. 135° 135º = 135º = ( 3π 4 π radians ) 180º radians ( –50° = –50° 6. –50° = 7. 5π 4 ( 5π 4 = – 5π 18 π radians ) 180º radians )( ) 5π 180º radians π radians 4 = 225º 8. π 10 π 10 = ( = 18º π 10 )( radians ) 180º π radians Sectors of Circles A sector is a region of a circle that is bounded by two radii and an arc of the circle. The central angle of a sector is the angle formed by the two radii. Arc Length and Area of a Sector The arc length s and area A of a sector with radius r and central angle (measures in radians) are as follows: Arc length: = Area: = Softball A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field. SOLUTION STEP 1 Convert the measure of the central angle to radians. π π radians ) 90º = 90º radians = 2 180º STEP 2 Find the arc length and the area of the sector. Arc length: s = r θ= 180 ( π )= 90π ≈ 283 feet 2 1 Area: A = 1 r2θ = (180)2 ( π ) = 8100π ≈ 25,400 ft2 2 2 2 ( Arc length: s = r θ= 180 ( π )= 90π ≈ 283 feet 2 1 Area: A = 1 r2θ = (180)2 ( π ) = 8100π ≈ 25,400 ft2 2 2 2 ANSWER The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet. 9.2 Assignment Page 566, 3-37 odd