Download Section 8A – Angles and Circles

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Section 8A – Angles and Circles
Angles
Definition.
• Given two rays sharing a common endpoint, the amount of rotation
about this common endpoint to move from one ray to the other is called an angle.
• The common endpoint is called the vertex.
• The side where the rotation begins is called the initial side or the initial position, and the final side is called the terminal side.
• If the rotation is counter-clockwise, then the angle is positive. If the rotation is
clockwise, then the angle is negative.
Definition. An angle is in standard position if it has its vertex at the origin and its
initial side along the positive x axis.
Definition. If an angle is in standard position, the reference angle is the smallest
angle between the terminal side and the x-axis.
Reference Angle
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Angle Measure
The size of an angle may be measured in revolutions (rev), in degrees (◦ ), or in radians
(rad). If no units are given, the angle is in radians. Radians are the unit of choice
in calculus.
Some common angles are:
Revolutions
Degrees
Radians
1 rev
360◦
2π
1
2
rev
180◦
π
1
4
rev
90◦
π
2
1
6
rev
60◦
π
3
1
8
rev
45◦
π
2
1
12
rev
30◦
π
6
Theorem. To convert an angle measure in one unit to another, multiply by the following. These all use the fact that 1 rev= 360◦ = 2π (Divide by the unit you have, and
multiply by the unit you want.)
• Revolutions to Radians:
2π
= 2π
1
• Revolutions to Degrees:
360◦
= 360◦
1
• Degrees to Revolutions:
1
360◦
• Degrees to Radians:
2π
π
=
360◦
180◦
• Radians to Degrees:
180◦
π
• Radians to Revolutions:
Example 1. Convert
radians.
7π
6
1
2π
into revolutions and radians. Also, give its reference angle in
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Example 2. Convert 120◦ into radians and revolutions. Also, give its reference angle
in radians.
Example 3. Convert
angle in radians.
7
8
revolutions into radians and degrees. Also, give its reference
Example 4. Convert the following angles from degrees into radians.
Definition.
90◦ ).
• An acute angle has measure between 0 and
• An obtuse angle has measure between
π
2
π
2
(or between 0 and
and π (or between 90◦ and 180◦ ).
• The complement of an angle whose measure is θ is an angle whose measure is
π
2 − θ.
• The supplement of an angle whose measure is θ is an angle whose measure is
π − θ.
Example 5. Sketch and label an obtuse angle θ in standard position and its supplement
α.
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Parts of a Circle
Definition. The set of all points P (x, y) in the plane that are a fixed distance, called
the radius r, from a fixed point, called the center C(h, k), is a circle, whose equation
has the standard form (x − h)2 + (y − k)2 = r2 .
• A radial line or radius is a line segment from the center of the circle to any
point on the circle. The word radius can refer to either a radial line or its length
r.
• A diameter line or diameter is any line segment between two points on the
circle which passes through the center. The word diameter can refer to either a
diameter line or its length, labeled d. Also, d = 2r.
• The circumference of the circle is the distance around the circle. The circumference is C = πd = 2πr.
• The area of the circle is A = πr2 .
• An arc is any piece of the circle between two points on the circle.
• A chord is any line segement between two points on the circle.
• A sector is any piece of the disk between two radial lines.
Definition. A central angle is an angle whose vertex is at the center of the circle.
The sides of a central angle are radial lines which intersect the circle at two points. the
arc between these points, the chord between these points, and the sector between the
two sides of the angle, are called the arc, chord, and sector subtended by the central
angle.
Theorem. For a circle of radius r, define the following notation:
C = circumference of the circle
L = length of an arc of the circle
A◦ = Area of the whole circle
A^ = area of a sector of the circle.
θ
θ
The fraction of the circle subtended by a central angle θ is 360
◦ for θ in degrees or 2π
for θ in radians. The arclength and area of the sector subtended by the angle theta are:
θ
C, for θ in radians
2π
θ
L=
C, for θ in degrees
360◦
L=
θ
A◦ , for θ in radians
2π
θ
A^ =
A◦ , for θ in degrees
360◦
A^ =
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Example 6. If a circle has an 16 mm diameter, find the arc length and the area of the
sector subtended by the central angle of 7π
4 .
Example 7. If a circle has an 28 mm diameter, find the arc length and the area of the
sector subtended by the central angle of 2π
3 .
Example 8. If a circle has a radius of length 1 unit, find the arc length of the sector
subtended by the central angle of 7π
6 .
Definition. The radian measure of an angle is defined to be the length of the sector
subtended by that angle when it is central angle for the unit circle. For a general circle
with radius r, the radian measure of an angle equals the arc length divided by the
radius r.
Definition.
• A secant line is a line that intersects the circle twice or is a line
that contains a chord of the circle.
• A tangent line is a line which intersects the circle at exactly one point, called
the point of tangency.