9.2 define general angles and use radian measure
DESCRIPTION
9.2 Define General Angles and Use Radian Measure. What are angles in standard position? What is radian measure?. Angles in Standard Position. - PowerPoint PPT PresentationTRANSCRIPT
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 radians180ºπ
12–= 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
( )12
12
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
( )12
12
9.2 Assignment
Page 566, 3-37 odd