When working with trigonometrical functions we often find the need to convert an angle in degrees to radians and vise verse.

Recall: 1 radian = degrees = 57.3°

To convert x° to radians we have:

radians

and to covert x radians o degrees we have:

x radians °

Fig 14-1

Example: 14.8° = 0.258 radians (Fig. 14-1)

Note:

1. The above setting of the Slide Rule would also give us 1.48° = 0.025 radians, 148° = 2.58 radians, etc. This is because the relationship between degrees and radians is a simple linear one. (e.g. 57.3° = 1 radian 114.6° = 2 radians, etc.)
2. For the C and D scales set as above (from steps 1 and 2), if the angle in degrees on the C scale takes us off the end of the D scale, we find the angle in degrees on the CF scale and hence its value in radians on the DF scale. For example, observe on Fig. 14-1 under the hair line, 46.5° on the CD scale gives 0.81 radians on the DF scale.
3. To convert from radians to degrees, we find the radian value on the D or DF scale and its degree equivalent is thus read off the C or CF scale.

Exercise 14(a)

Convert to Radians:

1. 24° =
2. 53°30’ =
3. 69°24’ =
4. 125° =
5. 1.25° =
6. 250° =

Convert to Degrees:

7. 1 radian =
8. 0.8 radians =
9. 3.8 radians =
10. 0.065 radians =
11. 6.5 radians =
12. 1.3 radians =

For small angles (i.e. below about 5° or 6°), the sine, tangent and radian value of an angle are all same to at least three figures. Thus, for an angle in degrees on the ST scale, its radians equivalent is read directly off the D scale. The actual graduations on the ST scale are only from 0.574° to 5.74°, 57.4° to 574°, 0.0574 to 0.574°, etc. This is because we have a linear relationship between degrees and radians, which is of course not so for an angle in degrees and its sine, cosine, tangent, etc.

Example 75° = 1.31 radians (Fig 14-2)

1. Set the hair line over 75 (marked as 0.75) on the ST scale.
2. Under the hair line read off 1.31 on the D scale as the answer.

Fig 14-2

Note:

1. In the above example we could take the same setting on the ST scale as 7.5°, 750°, 0.75°, etc., in which case the equivalent value in radians would have been 0.131, 13.1, 0.0131, etc. respectively.
2. To convert from radians to degrees, we find the radians value on the D scale and its degrees equivalent is read off the ST scale.

Covert to radians:

1. 4.5° =
2. 36° =
3. 360° =
4. 0.36° =
5. 91.36° =
6. 12’ =

Convert to degrees:

7. 0.65 radians =
8. 1.2 radians =
9. 0.12 radians =
10. 5 radians =
11. 0.625 radians =
12. 0.0314 radians =

Recall the formulae:

Arc Length, L = rθ (for r= radians and θ angle in radians)

Thus, if the angle θ was given in degrees (instead of radians), we could still evaluate the above quite readily.

Example: L = rθ for r = 14 and θ = 60° (Fig. 14-3)

Note: The area of a sector could be similarly found but would necessitate a second movement of the slide to multiply by the extra factor of r and the ½. A better method of obtaining the area of a sector is found in Unit 20.

Find the arc length given: