# 9.2 Define General Angles and Use Radian Measure What are angles in standard position? What is radian measure?

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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 can be formed by fixing one ray called the initial side and rotating the other ray called the terminal side, about the vertex. An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis. 0° 90° 180° 270° vertex 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.

Draw an angle with the given measure in standard position. SOLUTION a. 240º 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. SOLUTION b. 500º 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. SOLUTION c. –50º 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 a. –45º + 360º –45º – 360º There are many such angles, depending on what multiple of 360º is added or subtracted. = 315º = – 405º

Find one positive angle and one negative angle that are coterminal with ( b ) 395º. b. 395º – 360º 395º – 2(360º) = 35º = –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 65º – 360º 2. 230° 230º + 360º 230º – 360º = 425º = –295º = 590º = –130º

3. 300° 300º + 360º 300º – 360º 4. 740° 740º – 2(360º) 740º – 3(360º) = 660º = –60º = 20º = –340º

Degree and Radian Measures of Special Angles

a. 125º Convert ( a ) 125º to radians and ( b ) – radians to degrees. π 12 25π 36 = radians b. π 12 – π radians 180º π 12 – = radians () () = –15º ( π radians 180º ) = 125º

Convert the degree measure to radians or the radian measure to degrees. 5. 135° 135º 3π 4 = radians ( π radians 180º ) = 135º

6. –50° – 5 π 18 = radians 5π 4 = 225º π radians 180º 5π 4 = radians () () ( π radians 180º ) = –50° 7. 5π 4 8. π 10 = 18º π radians 180º π 10 = radians () () π 10

Sectors of Circles

Arc Length and Area of a Sector

A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field. Softball SOLUTION STEP 1Convert the measure of the central angle to radians. 90º = 90º ( π radians 180º ) = π 2 radians STEP 2 Find the arc length and the area of the sector. π Arc length: s = r = 180 = 90π ≈ 283 feet θ 2 ( ) Area: A = r 2 θ = (180) 2 = 8100π ≈ 25,400 ft 2 π 2 ( ) 1 2 1 2

The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet. ANSWER π Arc length: s = r = 180 = 90π ≈ 283 feet θ 2 ( ) Area: A = r 2 θ = (180) 2 = 8100π ≈ 25,400 ft 2 π 2 ( ) 1 2 1 2

9.2 Assignment Page 566, 3-37 odd