70 CALCULATOR FUNCTIONS AND FACILITIES
279. FUNCTIONS ON A SCIENTIFIC CALCULATOR RELEVANT TO NAVIGATION
Since calculators vary considerably in their manner of operation, some using the algebraic and some the Reverse Polish Notation, with very considerable differences in the methods of keying in entries, we have confined ourselves in this Manual for the most part to stating the formulae designed to solve the various problems and left the keying procedure to students, on the assumption that they are familiar with the operation of their own calculators.
The functions of scientific calculator keys can be somewhat intimidating at first. Perhaps an explanation of what the mnemonics mean will take away some of the mystery.
The keys are loosely grouped for reasons that seemed logical to the manufacturer. The numbers are located in the lower 4 or 5 row numeric pad. The numeric pad also contains the decimal point key, a key for changing a number’s algebraic sign, and key for entering powers of 10, the 4 arithmetic operators, and either an or key. The remainder of the keys are for manipulation of numbers and for the other mathematical operations. Sometimes these functions are accessed by pressing a shift key and then another key.
The key sequences required to obtain particular functions, or for operations such as the transfer of displayed values to the calculator memory, vary from machine to machine even when these are made by the same manufacturer. The typical procedures given in this Manual illustrate the principles involved, but the manufacturer’s handbook should be consulted for each model of calculator.
The Data Entry keys are used in entering, removing, and manipulating data to be used in subsequent calculations. The digit keys 0 to 9 and the decimal point key allow any number to be entered into the display in a logical left-to-right order. The calculator operates with a floating decimal point which can be placed wherever needed. A zero precedes the decimal point for numbers less than 1. Zeros trailing the last significant digit on the right of the decimal point are not displayed.
Pressing the change sign key instructs the calculator to change the sign of the displayed value. This allows the use of negative numbers in calculations. Some calculators have a negative key (-) to enter a negative value.
The basic arithmetic operations of addition, subtraction, multiplication and division are performed with 4 keys marked: , , , . On an calculator there will be an ENTER key to complete data entries, and on an algebraic calculator there will be a key marked either or to complete all pending operations and prepare the calculator for new calculations.
A function of a displayed number is, in most cases, obtained by depressing the relevant function key or, in appropriate cases, by depressing the second function key followed by the function key itself. The displayed number can be obtained either by entry from the key pad or as the result of a prior calculation.
Functions obtainable in this way include square, square root, inverse, common logarithm, natural logarithm and the trigonometrical functions. With the Equation Operating System, certain functions are entered before their argument. These include logarithms, trigonometric functions and negation of an expression. For example, with an EOS, log 54, cos. 23° and -246 would be entered as they are written.
Calculators handle a variety of calculations involving angles, for which the three common units for angular measure can usually be selected:
Degrees are each equal to I ÷ 360 of a circle. A right angle equals 90°
Radians are each equal to 1 ÷ 2πTr of a circle, equivalent to 57°.3. A right angle equals π ÷ 2 radians.
Gradients (usually abbreviated to Grads) are each equal to 1 ÷ 400 of a circle, equivalent to 0°.9. A right angle equals 100 Grads.
Radians and Gradients are not used in Navigation.
Most calculators are in the Degree Mode when turned on, this being indicated by DEG in the display, and this mode is used for all navigational calculations. Follow your calculator instructions if you wish to change the mode to RAD(ian) or GRAD(ient). It is important to make sure that the calculator is in the correct mode before commencing calculations with angles.
Most navigational calculations with angles involve degrees, minutes and seconds (sexagesimal measure) while calculators can only work in decimal measure. Sexagesimal to decimal conversion is described later in this Chapter.
The trigonometrical function keys , calculate the sine, cosine and tangent of the angle in the display. These trigonometrical functions relate the angles and sides of a right-angled triangle. To use these keys you enter the size of the angle, say 50°, then press the ‘tan’ , ‘cos.’ or ‘sin’ key thus:
As previously mentioned, if your calculator has an EOS, the function key would be pressed before the angle value. Where it is required to convert a tangent, sine or cosine into its corresponding angle, this is done by pressing the Inverse key and then the appropriate angle function key. For example, if you wish to find the angle corresponding to sine 0.7071068, you would press
.7071068 which would display 45°
Some calculators have a key labelled which you would use with the sin, cos. and tan keys instead of the inverse key.
Only sine’s, cosine’s and tangent’s are normally given on a calculator but these are invariably accompanied b a key which gives the reciprocal of any quantity usually labelled or
Since the cosecant of an angle is the reciprocal of its sine, the secant of an angle is the reciprocal of its cosine and the cotangent of an angle is the reciprocal of its tangent, these values would be found as follows:
Cosecant 50° = 50 (= 1.30541)
Secant 50° = 50 (= 1.55572)
Cotangent 50° = 50 (= 0.83910)
When it is required to convert a cosecant, secant or cotangent into its corresponding angle, this is achieved by pressing the reciprocal key first, followed by the inverse of the sine, cosine or tangent as appropriate.
The angle corresponding to cosecant 1.41421 = 1.41421 (=45°)
The angle corresponding to secant 1.22077 = 1.22077 (=35°)
The angle corresponding to cotangent 2.14451 = 2.14451 (=25°)
280. SEXAGESIMAL TO DECIMAL CONVERSION
For most navigational requirements angles must be expressed in degrees, minutes and seconds (or, more commonly, minutes and tenths of minutes), whereas angles are entered and presented in calculator displays in degrees and decimals of a degree. Some means of conversion from one to the other is very convenient and some, but not all, calculators are equipped with a key to perform this operation, so that degrees, minutes and seconds are entered and the conversion key pressed before the function required.
This degree/minute/second format can be expressed DDDMMSS, where DDD represents the whole angle, MM represents minutes and SS denotes seconds. To convert from the degree, minute, second format to the decimal degrees required by the calculator, enter the angle in the display as DDDMMSS and press the appropriate conversion key. This conversion key varies from calculator to calculator and the manufacturer’s handbook should be consulted.
As previously mentioned, it is convenient for many problems to utilise the hybrid sexagesimal/decimal notation of degrees, minutes and decimal fraction minutes rather than the full sexagesimal. In this case the decimal fraction minutes may have to be changed to seconds by multiplying by 6 prior to entry into the calculator display.
What is the tangent of 47° 36.4? The form of entry is usually DDDMMSS so that with an angle such as that under consideration the 0.4 may have to be turned into seconds.
47° 36.4 or 47° 36’24″ is entered into the calculator.
Depression of the tan key produces the answer 1.095395675 form 47.3624.
There will usually be some arrangement whereby the reverse of this operation can be performed so that an angle expressed in degrees, minutes and seconds can be extracted from the machine. The sexagesimal / decimal conversion process described above can be used to convert hours, minutes and seconds to decimal hours and vice versa.
281. MEMORY FACILITIES
Memory keys enable a displayed number to be transferred to a data memory or store for recall and later use during a calculation. Most calculators have at least one memory where you can store a number during calculation.
M + This key puts the number in the display into the memory. If there is a number already in the memory, the new number is added to it.
MS Some calculators have an MS or ‘Memory Store’ key. This puts, the number in the display into the memory and erases the number already there.
M – This subtracts the number in the display from the number in the memory.
MR This is the memory recall key. It puts the number in the memory into the display so that you can use it in the calculation you are doing. The number is still stored in the memory, however, so you can use it again if necessary.
MC The memory clear key erases a number from the memory. Some calculators have no memory clear key and so to clear the memory you press MR followed by M – or, on some calculators, the ordinary clear key, followed by MS
With scientific calculators having multiple memories (fiq. 70-1) the value shown in the display is transferred to memory with a key marked , and calculators of this type usually require the depression of an alphanumeric key in addition to the key to select a particular data store, e.g. (or A to transfer to memory store number (or store A).
The m key recalls to the display the number in the data memory ‘m’. For instance, the key sequence recalls to the display the number that was in data memory number 2. The number that was in the display is lost.
The m key exchanges the value in the display with the value in the data memory ‘m’. For instance, the key sequence stores the value 3 in data memory and displays the value that was in data memory
The results of calculations may be stored in a data memory by entering a value, pressing , entering the operation to be performed and entering the number of the data memory in which to store the result. These key sequences are used to accumulate results from a series of independent calculations. The displayed number and calculations in progress are not affected.
To use these sequences: –
- enter the number that is to operate on the memory value
- press STO
- enter the operation to be performed
- enter the number of the memory to be used
The use of memory arithmetic in this way is best described by an example:
Find the total of the three separate computations below: (fig: 70-2)
28.3 x 7 = 198.1
173 + 16 = 189
31 – 42 + 7.8 = -3.2
TOTAL = 383.9
282. POLAR, RECTANGULAR CONVERSION
The facility of Polar/Rectangular conversion is one of the most important features of a scientific calculator which is to be put to navigational use.
The polar system of co-ordinates describes a point in terms of a line drawn from a centre to the point using two numbers, the first being the length of the line (labelled r in fig. 70-3) and the second the number of degrees the line is from the vertical [labelled theta (θ°) in fig. 70-3]. Navigators will immediately recognise these co-ordinates as the Course (θ°) and the Distance (r) between two points, in this case between points 0 and P.
The rectangular co-ordinate system describes a point in terms of its distance from the y axis (OY in fig. 70-3), the distance being BP which navigators will recognise as the difference of Lat. (d Lat); and with its distance from the x axis (OX in fig. 70-3), this distance being AP which is called the departure (Dep.). The conversion from polar to rectangular co-ordinates (or back from rectangular co-ordinates to polar co-ordinates) can be performed instantly on a scientific calculator possessing the Polar/Rectangular conversion feature.
By entering the Course and Distance and bringing into action the key the D Lat, and Dep. are calculated. Some calculators of this type will also have on the keyboard an or an key.
This renders interchangeable the display (called the x – Register) with another register (referred to as the y or t – Register). The key operates on the numbers placed in these two registers.
The order in which D Lat and Dep. or Course and Distance are entered and displayed will depend upon the make of the machine. We will assume, for illustrative purposes, that the Course or the Dep. is displayed and the Distance or D Lat is buried.
For example, suppose you wished to find the position arrived at after sailing for a distance of 23 miles on Course 247° T from a position in Lat 42° 21.0 N, Long 17° 43.0 W. The D Lat and Dep. could be found readily using the Polar/Rectangular conversion keys. The procedure varies slightly from calculator to calculator so the manufacturer’s instructions should be consulted for converting polar to rectangular co-ordinates. Following the recommended procedure, enter the distance (23) and the Course (247°) and the calculator will produce the D Lat – 8.9868 (the minus sign indicates that the D Lat is S. – ) and the accuracy is superfluous so we can say D Lat = 9′ S. The calculator will also produce Dep. -. 21.171612 (in this case the minus sign indicates the departure is W.) and we can say the Dep. = 21′.2 W.
The Mean Lat. would be 42° 21.0 – 4.5 = 42° 16.5. The departure is converted into D Long thus: –
The position arrived at would be: –
On some advanced scientific calculators the or registers are replace by internal registers separated by a comma, both registers being accessed by operation of a cursor key.
When it is required to convert rectangular co-ordinates to polar, the manufacturer’s instructions should again be consulted for the correct procedure. In all these conversions it is necessary to know the conventions for the use of angles, and positive/negative signs for direction.
Fig. 70-4 shows the plane triangles for Courses in 3-figure notation (denoted ‘C’) and in quadrantal notation (denoted ‘Q’) for each of the four quadrants. In each case, C (or Q) and D (the distance) are the polar co-ordinates of the terminal point of the track from the start position, and the D Lat and Dep. are the rectangular coordinates.
The units and sign convention used in all calculations must be consistent, i.e., D (the distance) in miles and D Lat in minutes of arc. Angle C is always in decimal degrees clockwise notation. The sign convention for Lat and Dep. is as shown in fig. 70-4 i.e.,
Lat N and Long E + ve (positive) Lat S and Long W – ve (negative)
Now suppose it is required to find the Course and Distance where the D Lat is 2°18.4 N. and the Dep. is 256.3 W between the initial and final positions. For use in the calculator this D.Lat would be 138.4, and a calculator with registers would be used as follows: –
Following the maker’s instructions for Rectangular to Polar conversion, enter the D Lat (138.4) and the Dep. (-256.3). This will produce a distance of 291.2804 and an angle of – 61.6313.
The departure was entered negative ( – 256.3) following the convention that W. is negative, and the negative sign for the angle indicates that the angle is measured anticlockwise from N., hence 360° is added to give the required Course ( – 61.63 + 360) = 298°.4. The required Course and Distance is therefore Course 298° T and Distance 291.3 miles.