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) =  / 7) /π.

                From the 1st and 3rd ratios: D = 180X / 7) / π  = 180 / 7  = 25.7 degrees.

                From the 2nd and 3rd ratios: G = 200X / 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?