Chapter 1

The Units of Arc and Angle

Definition of Angle: Angle is defined as the
space between two crossing lines, as shown below**:**

We may call an angle a "central angle" if its vertex is at the
center of a circle, as shown**.** The three angles shown are central
angles. Anglewise**,** the size of any
central angle is equal to its
opposite arc as shown below**:**

Three different units are defined for the measurement of arc or angle**:**
degree, grad, and radian.

1) __Degree__

1/360 of a circle is defined as 1° called "one degree."

1/60 of a degree is 1 minute,
and 1/60 of a minute is 1 second of angle**.**

2) __Grad__

1/400 of a circle is defined as 1 grad.

3. __Radian__

One radian of angle is the central angle in any circle
which opposite arc is equal to the radius of that circle. If a
piece of string is cut equal to the radius R of a circle and then placed on the
edge of that circle, as shown, the central angle corresponding (or opposite) to that arc is
called one "radian**.**"

Example 1: Naming 3**.**14rd as "
π ", calculate
angles 360°, 180°,
90°, 60°, 45°, and 30° in terms of
π.
The pronunciation of the symbol is "pi."

**Solution: **

Fraction: | 1 full circle | 1/2 circle | 1/4 circle | 1/6 circle | 1/8 circle | 1/12 circle |

Degrees: | 360° | 180° | 90° | 60° | 45° | 30° |

Radians: | 2π = 6.28rd | π | π/2 | π/3 | π/4 | π/6 |

Conversion Between Units of Angle:

To convert from Degrees D to grad **G** or to
Radians rd, the following relations apply**:**

Example 2: Convert D = 18° to grads and radians.

**Solution: **According to the
above formula, we may write**:** (18/180) = (G/200)
= (rd/3**.**14).

From the first two ratios**:** G = (200X18) / 180 =
20 grads.

From the 1st and 3rd ratios**:** rd = (18X3**.**14) / 180 =
0.314 radians.

Example 3: Convert G = 15 grads to degrees and radians.

**Solution: **According to the
above formula, we may write**:** (D/180) = (15/200)
= (rd/3**.**14).

From the first two ratios: D = (180X15) / 200 = 13.5 degrees.

From the last two ratios: rd = (15X3**.**14) / 200 =
0.236 radians.

Example 4: Convert rd = π/7 radians to degrees and grads.

**Solution: **According to the
above formula, we may write**:** (D/180) = (G/200)
= (π /
7) **/**π.

From the 1st and 3rd ratios**: ** D = 180X(π /
7) / π =
180 / 7 = 25**.**7 degrees.

From the 2nd and 3rd ratios**:** G = 200X(π /
7) / π =
200 / 7 = 28**.**6 grads.

__Arc Length-Central Angle Formula:__

There is an easy formula that relates any central angle (
*θ* ) to its opposite arclength (
s )
and the radius ( R ) of a circle. This formula is valid only if the central angle is
measured or expressed in radians**.**

Example 5: Referring
to the above figure, suppose angle
θ
is 148° and R = 1**.**25 in.
Calculate the length of arc AB**.**

**Solution: **Using** **
**S = R**
θ, and converting degrees to radians, yields:
S = (1**.**25 in.)(148°)**( **
3.14rd / 180°** )** = 3.2 in.

__Exercises:__

1) Convert the following angles from radians to degrees and
grads**:** a) π/2
b)π c) 3π/2**.**

2) Convert G = 58**.**642 grads to degrees and radians**.**

3) Convert 168°, 32 min**.**
to grads and radians**.**

4) Convert 3π/11
from radians to degrees and grads**.**

5) Convert each of 120° and
225° to radians**.**

6) The sum of 3 angles is 150grads and their relative sizes are
proportional to numbers 2,3, and 4**.** Find the value of each in terms of
degrees and radians**.**

7)Determine the angle (in radians) that the minute indicator of a clock sweeps during a 48 minute time interval.

8) In a circle with a radius of 2**.**50cm**, **find the
central angle (in radian) that is opposite to an arc of length 8**.**75cm.

9) What angle (in radians) do the hour and minute indicators of a clock make with each other at 24minutes past Noon?