9.2 Define General Angles and Use Radian Measure

What are angles in standard position?What is radian measure?

Angles in Standard PositionIn 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

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 radians or 180°= radians.

Converting Between Degrees and Radians

Degrees to radians

Multiply degree measureby

Radians to degrees

Multiply radian measure by

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. All other special angles are multiples of these angles.

a. 125º

Convert (a) 125º to radians and (b) – radians to degrees.

π12

25π36= radians

b.π12

–π radians

180ºπ12

–= radians( ) ( )

= –15º

( π radians180º

)= 125º

Convert the degree measure to radians or the radian measure to degrees.

5. 135°

135º

3π4= radians

( π radians180º

)= 135º

6. –50° –50°

– 5π18= radians

5π4

= 225º

π radians180º5π

4= radians( ) ( )

( π radians180º

)= –50°

7. 5π4

8. π10

= 18º

π radians180ºπ

10= radians( ) ( )π

10

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 SectorThe arc length s and area A of a sector with radius r and central angle (measures in radians) are as follows:

Arc length:

Area:

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 1 Convert the measure of the central angle to radians.

90º = 90º( π radians

180º

)=

π2

radians

STEP 2 Find the arc length and the area of the sector.πArc length: s = r = 180 = 90π ≈ 283 feetθ 2

( )

Area: A = r2θ = (180)2 = 8100π ≈ 25,400 ft2π2

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The length of the outfield fence is about 283 feet. The area of the field is about 25,400 square feet.

ANSWER

πArc length: s = r = 180 = 90π ≈ 283 feetθ 2( )

Area: A = r2θ = (180)2 = 8100π ≈ 25,400 ft2π2

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9.2 Assignment

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