Transcript 9.2 Define General Angles and Use Radian Measure
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.
180° 90°
vertex
An angle is in standard position if its vertex is at the origin and its initial side lies on the positive 270° 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 – = –60º
4.
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 𝜋𝑟, there are 2 𝜋 radians in a full circle.
Degree measure and radian measure are related by the equation 360 ° = 2𝜋 radians or 180°= 𝜋 radians.
Converting Between Degrees and Radians
Degrees to radians Multiply degree measure by 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 180° Radians to degrees Multiply radian 180° measure by 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
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 𝜋 radians).
It will be helpful to memorize the equivalent degree and radian measures of special angles in the 1 st for 90° = 𝜋 2 quadrant and radians. All other special angles are multiples of these angles.
Convert (
a
)
125º
degrees.
a. 125º = 125º (
to radians and ( b )
– π
radians
180º ) = 25π 36
radians
π 12
radians to b.
– π 12 = ( – π 12
radians
) ( 180º π
radians
) = –15º
Convert the degree measure to radians or the radian measure to degrees.
5. 135° 135º = 135º ( π
radians
180º ) = 3π 4
radians
6.
–50° –50° = = –50° ( π
radians
180º ) – 5 π 18
radians 7.
8.
π 5π 4 10 π 10 5π 4 = ( = 225º 5π 4
radians
) ( 180º π
radians
) = ( π 10
radians
) ( 180º π
radians
) = 18º
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
𝜃 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 𝜃 (measures in radians) are as follows:
Arc length:
𝒔 = 𝒓𝜽
Area:
𝑨 = 𝟏 𝟐 𝒓 𝟐 𝜽
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.
90º = 90º ( π
radians
180º ) = π 2
radians STEP 2 Find the arc length and the area of the sector.
Arc length:
s = r
Area:
A = 1 2
θ
( π 2
r
2 θ = (180) 2 2 ) ( π 2 ) feet = 8100π ≈ 25,400 ft 2
Arc length: Area:
A = s = r 1 2
r
2
θ
2 ( π 2 ) feet θ = (180) 2 ( π 2 ) = 8100π ≈ 25,400 ft 2
ANSWER The length of the outfield fence is about
283
The area of the field is about
25,400 feet square feet
.
.