INSTRUCTION LEAFLET
for the
FEARNS TRIGONOMETRY CALCULATOR

The Fearns Trigonometry Calculator consists of; a main dial; a smaller top dial and a cursor to facilitate interpolating between graduations.
 
‘Y’ scale - This Is the outer red scale on the main dial. It is a simple log scale with a range (including the overlap portion) from .001 to 100.
‘R’ or X’ scale — This is the black scale on the periphery of the top dial. It also is a simple log scale, and has a range from .1 to 100. It is used for values of ‘R’ or values of ‘X’. If the calculation in question involves values of both ‘R’ and ‘X’, then use this scale for the ‘R’ value and the ‘Y’ scale for the ‘X’ value.
Angle scale — There are, in addition to the above, four angle scales, two of which show in each of the two windows. Each scale is calibrated in degrees and minutes, and the spacing of the graduations on each respectivelyis proportional to log.cosine a; log.sine a; log.cotangent a ;and log.tangent a , as indicated on the top dial, and where a is the angle inquestion. For ease of reference, the window on the left-hand side of the top dial (when the latter is the correct way up) will be referred to as the left-hand window, and the other as the right hand window. Thus, the cosine and sine graduations are in the left-hand window, and the cotan.and tan. graduations in the Right-hand window.
 
Nomenclature — The following symbols are used throughout:—
X = Base of triangle
Y = Vertical height of triangle
R = Hypotenuse of triangle
Note: The angle between X and Y is always 90° , and the other two angles are always the ones concerned in the calculations.

Trigonometrical Functions

To find the Cosine of an angle — set the upper graduation on the left-hand window (i.e. the one marked “Cosine”) to the angle, and read off the cosine on scale “Y” opposite the black arrow.
 
To find the Sine, Tangent, or Cotangent of an angle — proceed exactly as above, but setting the appropriate graduation to the angle. (e.g. If tangent is required, set the graduation marked “Tan.” to the angle.)
 
EXAMPLES — Find the cosine of 75° — 30’.
Set the Cosine graduation to 75° — 30’ and read the
required value of .250 opposite black arrow.
  Find the tangent of 19°.
Proceed as before, but set the “Tan.” graduation to the
angle.
Answer .345
Note: Use the values on the outer portion of the Y scale, for Sines and Tangents if the angle is below 6°, and for Cotangents and Cosines if the angle is over 84°.
 
EXAMPLE — Find the Cotangent of 86° — 10'.
Answer (on outer scale) .0670
 
To find an angle when its cosine, sine, tangent, or cotangent is known — use the same scales as before but reverse the procedure. i.e. Set the black arrow to the known value of the function and read off the angle in the window opposite the appropriate graduation.
 
EXAMPLE — Find the angle whose sine is .380
Set the black arrow to .380 on scale Y, and read the answer
22° — 20’ in the left-hand window opposite the “Sine”
graduation.
To find a function of an angle, given another function of the same angle —set the black arrow to the value of the known function on scale Y and read the angle in the appropriate window. Re-set same angle for the required function, and read answer opposite black arrow.
 
EXAMPLE — Find the value of Sin (Tan-1  .550).
Set black arrow to .550 and read angle opposite “Tan” graduation, i.e. 28° — 50’. Then set “Sine” graduation to 28° — 50’ and read the required answer .480 opposite black arrow.
 

Solution of Right-Angled Triangles
 

General — The procedure for solving any right-angled triangle is very simple. Normally in any given problem there are two known values and one unknown or required value. Procedure is to set the calculator to the known values and read off the required value, taking care to use the correct window and scales, thus:
(a) It two sides are known — set one side against the other on the outer scales, and read off the angles in the appropriate window.
(b) If one side and one angle are known — set the appropriate graduation to the known angle; set the cursor over the known side on the appropriate outer scale and read off the value of the other side on the adjacent outer scale.
(c) If two sides are known and the other side is required — first determine the angles as in (a) and then determine the other side as in (b).
 
Note: As will be observed from the diagrams on the top dial, if the hypotenuse R is involved in the calculation, the left-hand window only is used for angle setting or reading, and if the base X and perpendicular Y only are involved use the right-hand window.
 
EXAMPLES —  
for (a) If X equals 2 and Y equals 5, find the angles.
Set 2 against 5 on the outer scales and read angles in right-hand window.
Answer 68° - 10' and 21° - 50'.
for (b) If R equals 7 and the included angle between R and Y equals 70° find Y.
Set the graduation on the top of the left-hand window to 20°, and set the cursor over 7 on the black outer scale.
Read off the answer Y equals 2.39
 

Using the Trigonometry Calculator as a Circular Slide Rule
for Multiplication and Division
 

As the Y scale and the R/X scales are simple logarithmic scales it will be readily appreciated that they can be used for multiplication and division as with a normal slide rule. Procedure is exactly the same as with a straight slide rule, the only difference being that the scales are circular.
 

Vectors. Cartesian and Polar Co-ordinates
 

The vector R or She point P in Co-ordinate Geometry is
expressed in Polar Co-ordinates as R = R,q  where the length R
is the modulus, radius vector, or numerical value, and q is the
argument or vectorial angle.
The same vector or point can also be expressed in its Cartesian Co-ordinates X and Y, and the following relationship between the two co-ordinate systems exists.
 
X = R. cos q Y = R. sin q
q = tan -1 (Y/X) R = Φ(X2 + Y2)
 

Conversions between Cartesian and Polar Co-ordinates
 

(a) To convert Cartesian Co-ordinates X and V to Polar Co-ordinates R and q.
First set X to V on outer scales and read q in right-hand window against Tan, graduation.
Then set q against sine graduation in left-hand window and against value of Y on outer scale read value of R.
 
EXAMPLE — Find the Polar Co-ordinates of 4 and 3.
Set 4 on X scale to 3 on Y scale and read 37° against Tan.
graduation in right-hand window. Set 37° against Sine
graduation in left-hand window and against 3 on Y read
5 on R scale.
Answer 4, 3 = 5, 37°
 
(b) To convert Polar Co-ordinates R & q to Cartesian Co-ordinates X and Y
First set q against Cosine graduation in left-hand window and against value of R on R scale read value of X on the outer red scale (using the Y scale for X values in this instance).
Then set q against Tan. in the right-hand window and against X on the X scale read value of Y on the Y scale.
 
EXAMPLE — Find the Cartesian (or Rectangular co-ordinates) of 2, 60°.
Set 60° against Cosine graduation in left-hand window and
against 2 on R scale read X = 1.0
Set 60° against Tan, graduation in right-hand window and
against 1.0 on X scale read Y= 1.73
Answers 2,60° = 1, 1.73
 

Complex Numbers
 

A complex number is represented similarly by a point P on an Argand Diagram in which the real part X and the imaginary part Y of the complex number are stated as:
R = X+jY
    = /R/[ cos q + j. sin q ]
    =  /R/ ejq
 
The real part and imaginary part of the complex number representing a vector R in the diagram can be determined as in (a) above. Conversely the numerical value or modulus R, and the argument or vectorial angle q of the complex number X + j Y can be determined as in (b) above.
 

Electrical A.C. Power Calculations
 

The Electrical Engineer will readily observe that the calculator can be used for solving A.C. power calculations in a circuit by using the following relationships:
 
R = Total Power in volt-amperes
X = Active Power in watts
Y = Reactive Power in reactive volt-amperes
cos = Power Factor = X/R

 

EXAMPLE —  Find the power factor (and angle) of a load which takes 15 volt-amperes and 12 watts. (i.e. We know R and X and require to know cos q and q, therefore use left-hand window and use the Y scale for X.)
Set 15 on R scale to 12 on Y scale (representing X) and read off Power Factor, cos q = 0.8 on Y scale opposite black arrow.
In left-hand window read off = 37° opposite cosine graduation.
 

FEARNS CALCULATORS
Atlas Works, West View Terrace,
Dunston, Gateshead NEl 1 9EL, England.
Telephone: Dunston 504086
 

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