angle of circle

Degree/Radian Circle

In everyone’s experience it is usual to measure angles in degrees. We learn early in childhood that there are 360 degrees in a circle, that there are 90 degrees in a right angle, and that the angle of an equilateral triangle contains 60 degrees. On the other hand, to scientists, engineers, and mathematicians it is usual to measure angles in radians.

The size of a radian is determined by the requirement that there are 2 radians in a circle. Thus 2 radians equals 360 degrees. This means that 1 radian = 180/ degrees, and 1 degree = /180 radians.

The reason for this is that so many formulas become much easier to write and to understand when radians are used to measure angles. A very good example is provided by the formula for the length of a circular arc. If A and B are two points on a circle of radius R and center C, then the length of the arc of the circle connecting them is given by

d(A,B) = R a,

where R is the radius of the sphere, and a is the angle ACB measured in radians. If we measure the angle in degrees, then the formula is

d(A,B) = R a/180,

These formulas can be checked by noticing that the arc length is proportional to the angle, and then checking the formula for the full circle, i.e., when a = 2 radians (or 360 degrees).

The figure below gives the relationship between degrees and radians for the most common angles in the unit circle measured in the counterclockwise direction from the point to the right of the vertex. The form of the ordered pair is {degree measure, radian measure}

In mathematics and physics, the radian is a unit of angle measure. It is the SI derived unit of angle. It is defined as the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. Angle measures in radians are often given without any explicit unit. When a unit is given, it is rad.

There are 2 π (about 6.283185) radians in a complete circle, so:
2π rad = 360°
1 rad = 360/(2π)° = 180/π° (approximately 57.29578°).

or:
360° = 2π rad
1° = 2π/360 rad = π/180 rad
In calculus, angles must be represented in radians in trigonometric functions, to make identities and results as simple and natural as possible. For example, the use of radians leads to the simple identity

which is the basis of many elegant identities in mathematics.

The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995.

Angle at the Circumference

If the end points of an arc are joined to a third point on the circumference of a circle, then an angle is formed.
For example, the minor arc AB subtends an angle of 45º at C.  The angle ACB is said to be the angle subtended by the minor arc AB (or simply arc AB) at C.

In the circle, the angle subtended by the minor arc AB at C is 45 degrees.

The angle ACB is an angle at the circumference standing on the arc AB.

Angle at the Centre

If the end points of an arc are joined to the centre of a circle, then an angle is formed.  For example, the minor arc AB subtends an angle of 105º at O.  The angle AOB is said to be the angle subtended by the minor arc AB (or simply arc AB) at the centre O.

The minor arc AB subtends the angle at the centre, AOB, of size 105 degrees.

The angle AOB is an angle at the centre O standing on the arc AB.

Angle at Centre Theorem

Theorem

Use the information given in the diagram to prove that the angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc.

In the circle, angle AOB at the centre is subtended by the arc AB and angle ACB at the circumference is also subtended by the arc AB.  The diameter of the circle passing through O and touching C forms angles of size a degrees and b degrees near the point C and x degrees and y degrees near the centre O.

Given:

Angle AOB and angle ACB stand on the same arc; and O is the centre of the circle.

To prove:

Angle AOB = 2(angle ACB)

Proof:

In the circle, angle AOB at the centre is subtended by the arc AB and angle ACB at the circumference is also subtended by the arc AB.  The diameter of the circle passing through O and touching C forms angles of size a degrees and b degrees near the point C and x degrees and y degrees near the centre O.  Angle CAO is a degrees and angle CBO is b degrees.

From triangle OAC, x = a + a  {Exterior angle of a triangle}.  So, x = 2a  ...(1).  From triangle OBC, y = b + b  {Exterior angle of a triangle}.  So, y = 2b  ...(2).  Adding (1) and (2) gives x + y = 2a + 2b = 2(a + b).  So, angle AOB = 2(angle ACB) as required.

In general:

The angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc. This is called the Angle at Centre Theorem.

We also call this the basic property, as the other angle properties of a circle can be derived from it.

Example 22

Find the value of the pronumeral in the following circle centred at O.

In the circle, the angle AOB at the centre subtended by arc AB is 4x degrees and the angle ACB at the circumference subtended by arc AB is 60 degrees.

Solution:

Example 23

Find the value of the pronumeral in the following circle centred at O.

Solution:

Angle in a Semi-Circle

In general:

The angle in a semi-circle is a right angle.

Example 24

Find the value of the pronumeral in the following circle centred at O.

Solution:

by,

Kong Ngiik Hiong

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