Write the expression in simplest form. 1. (180º) 5 6 ANSWER 150º 2. 115º 180º 36 ANSWER 23

Write the expression in simplest form. 11 12 3. (180º) ANSWER 165º 4. 135º 180º ANSWER 3 4

Write the expression in simplest form. 5. A compact disc has radius 6 centimeters. Find its circumference and area to the nearest tenth. ANSWER C 37.7 cm; A 131.1 cm2

EXAMPLE 1 Draw angles in standard position 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.

EXAMPLE 1 Draw angles in standard position 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.

EXAMPLE 1 Draw angles in standard position 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.

EXAMPLE 2 Find coterminal angles Find one positive angle and one negative angle that are coterminal with (a) –45º and (b) 395º. SOLUTION There are many such angles, depending on what multiple of 360º is added or subtracted. a. –45º + 360º = 315º –45º – 360º = – 405º

EXAMPLE 2 Find coterminal angles = –325º b. 395º – 360º = 35º 395º – 2(360º)

GUIDED PRACTICE for Examples 1 and 2 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

GUIDED PRACTICE for Examples 1 and 2 3. 300° 300º + 360º = 660º 300º – 360º = –60º 4. 740° 740º – 2(360º) = 20º 740º – 3(360º) = –340º

( ) ( ) EXAMPLE 3 Convert between degrees and radians Convert (a) 125º to radians and (b) – radians to degrees. π 12 ( π radians 180º ) = 125º a. 125º 25π 36 = radians b. π 12 – π radians 180º π 12 – = radians ( ) = –15º

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

( ) ( ) GUIDED PRACTICE for Example 3 6. –50° π radians –50° = –50° 180º ) = –50° –50° – 5π 18 = radians 7. 5π 4 5π 4 π radians 180º 5π 4 = radians ( ) = 225º

( ) GUIDED PRACTICE for Example 3 π 8. 10 π π 180º radians = 10 10 = 18º

EXAMPLE 4 Solve a multi-step problem Softball A softball field forms a sector with the dimensions shown. Find the length of the outfield fence and the area of the field.

( ) EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Convert the measure of the central angle to radians. 90º = 90º ( π radians 180º ) = π 2 radians

( ) ( ) EXAMPLE 4 Solve a multi-step problem 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 = r2θ = (180)2 = 8100π ≈ 25,400 ft2 π 2 ( ) 1 The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet. ANSWER

( ) GUIDED PRACTICE for Example 4 9. What If? In Example 4, estimate the length of the outfield fence and the area of the field if the outfield fence is 220 feet from home plate. SOLUTION STEP 1 Convert the measure of the central angle to radians. 90º = 90º ( π radians 180º ) radians = π 2

( ) ( ) GUIDED PRACTICE for Example 4 STEP 2 Find the arc length and the area of the sector. π Arc length: s = r = 220 = 110π ≈ 346 feet θ 2 ( ) Arc length: A = r2θ = (220)2 = 12100π ≈ 38,013 ft2 π 2 ( ) 1 The length of the outfield fence is about 346 feet. The area of the field is about 38,013 square feet. ANSWER

Daily Homework Quiz Find one positive angle and one negative angle that are conterminal with 475°. 1. 115°; –245° ANSWER 2. Convert 315° to radians and radians to degrees. 17π 6 ANSWER radians; 510° 7π 4

Daily Homework Quiz 3. You are planting a vegetable garden on a plot of land that is a sector of a circle. You want fencing along only the curved edge of the garden. Use the figure to find the length of fencing you will need and the area that will be available for planting. ANSWER about 19.6 ft; about 147 ft2