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99-458: Replacing the previously standard hand-held slide rule -type devices (known as the "banjo" and "is/was"), the TDC was designed to provide fire-control solutions for submarine torpedo firing against ships running on the surface (surface warships used a different computer). The TDC was a rather bulky addition to the sub's conning tower and required two extra crewmen: one as an expert in its maintenance,

198-427: A lookup table that maps from position on the ruler as each function's input. Calculations that can be reduced to simple addition or subtraction using those precomputed functions can be solved by aligning the two rulers and reading the approximate result. For example, a number to be multiplied on one logarithmic-scale ruler can be aligned with the start of another such ruler to sum their logarithms. Then by applying

297-409: A 10 cm (3.9 in) circular would have a maximum precision approximately equal to a 31.4 cm (12.4 in) ordinary slide rule. Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around"; they never have to be reoriented when results are near 1.0—the rule is always on scale. However, for non-cyclical non-spiral scales such as S, T, and LL's,

396-438: A calculation are generally done mentally or on paper, not on the slide rule. Most slide rules consist of three parts: Some slide rules ("duplex" models) have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip (which can usually be pulled out, flipped over and reinserted for convenience), still others on one side only ("simplex" rules). A sliding cursor with

495-662: A common ship speed. As with the angle solver, the equations implemented in the position keeper can be found in the Torpedo Data Computer manual. Similar functions were implemented in the rangekeepers for surface ship-based fire control systems. For a general discussion of the principles behind the position keeper, see Rangekeeper . Slide rule A slide rule is a hand -operated mechanical calculator consisting of slidable rulers for evaluating mathematical operations such as multiplication , division , exponents , roots , logarithms , and trigonometry . It

594-443: A free dish and one cursor. The dual cursor versions perform multiplication and division by holding a constant angle between the cursors as they are rotated around the dial. The onefold cursor version operates more like the standard slide rule through the appropriate alignment of the scales. The basic advantage of a circular slide rule is that the widest dimension of the tool was reduced by a factor of about 3 (i.e. by π ). For example,

693-537: A similar method. In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea , Asia Minor) gave the first tables of chords, analogous to modern tables of sine values , and used them to solve problems in trigonometry and spherical trigonometry . In

792-404: A simplified example). The TDC provided a continuous solution to this equation using data updates from the submarine's navigation sensors and the TDC's target tracker. The TDC was also able to automatically update all torpedo gyro angle settings simultaneously with a fire control solution, which improved the accuracy over systems that required manual updating of the torpedo's course. The TDC enables

891-528: A spread (i.e. multiple launches with slight angle offsets) to increase the probability of striking the target given the inaccuracies present in the measurement of angles, target range, target speed, torpedo track angle, and torpedo speed. Salvos and spreads were also launched to strike tough targets multiple times to ensure their destruction. The TDC supported the firing of torpedo salvos by allowing short time offsets between firings and torpedo spreads by adding small angle offsets to each torpedo's gyro angle. Before

990-495: A standard ruler , the slide rule is not meant to be used for measuring length or drawing straight lines. Nor is it designed for addition or subtraction, which is usually performed using other methods, like using an abacus . Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results. English mathematician and clergyman Reverend William Oughtred and others developed

1089-484: A submarine requires parallax and torpedo ballistic corrections when gyro angles are large. These corrections require knowing range accurately. When the target range was not known, torpedo launches requiring large gyro angles were not recommended. Equation 1 is frequently modified to substitute track angle for deflection angle (track angle is defined in Figure 2, θ Track = θ Bow + θ Deflection ). This modification

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1188-511: A target will be based on where it is now and was) or mechanical calculator/sights. These were often "woefully inaccurate", which helps explain why torpedo spreads were advised. During World War II, Germany, Japan, and the United States each developed analog computers to automate the process of computing the required torpedo course. In 1932, the Bureau of Ordnance (BuOrd) initiated development of

1287-399: A vertical alignment line is used to find corresponding points on scales that are not adjacent to each other or, in duplex models, are on the other side of the rule. The cursor can also record an intermediate result on any of the scales. Scales may be grouped in decades , where each decade corresponds to a range of numbers that spans a ratio of 10 (i.e. a range from 10 to 10 ). For example,

1386-749: Is De Triangulis by the 15th century German mathematician Regiomontanus , who was encouraged to write, and provided with a copy of the Almagest , by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond . Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by

1485-546: Is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies . The Greeks focused on the calculation of chords , while mathematicians in India created

1584-425: Is again positioned to start at the 2 on the bottom scale. Since 2 represents 20 , all numbers in that scale are multiplied by 10 . Thus, any answer in the second set of numbers is multiplied by 100 . Since 8.8 in the top scale represents 88 , the answer must additionally be multiplied by 10 . The answer directly reads 1.76 . Multiply by 100 and then by 10 to get the actual answer: 1,760 . In general,

1683-500: Is for base e. Logarithms to any other base can be calculated by reversing the procedure for calculating powers of a number. For example, log2 values can be determined by lining up either leftmost or rightmost 1 on the C scale with 2 on the LL2 scale, finding the number whose logarithm is to be calculated on the corresponding LL scale, and reading the log2 value on the C scale. Addition and subtraction aren't typically performed on slide rules, but

1782-404: Is illustrated with Equation 2 . where θ Track is the angle between the target ship's course and the torpedo's course. A number of publications state the optimum torpedo track angle as 110° for a Mk 14 (46 knot weapon). Figure 4 shows a plot of the deflection angle versus track angle when the gyro angle is 0° ( i.e. ., θ Deflection = θ Bearing ). Optimum track angle is defined as

1881-468: Is one of the simplest analog computers . Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule is closely related to nomograms used for application-specific computations. Though similar in name and appearance to

1980-613: Is possible using either of the following two techniques: Using (almost) any strictly monotonic scales , other calculations can also be made with one movement. For example, reciprocal scales can be used for the equality 1 x + 1 y = 1 z {\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {1}{z}}} (calculating parallel resistances , harmonic mean , etc.), and quadratic scales can be used to solve x 2 + y 2 = z 2 {\displaystyle x^{2}+y^{2}=z^{2}} . The width of

2079-573: Is the area of the triangle and R is the radius of the circumscribed circle of the triangle: The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: or equivalently: The law of tangents , developed by François Viète , is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables. It

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2178-653: Is to expand the letters into a sentence, such as " S ome O ld H ippie C aught A nother H ippie T rippin' O n A cid". Trigonometric ratios can also be represented using the unit circle , which is the circle of radius 1 centered at the origin in the plane. In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = cos ⁡ A {\displaystyle x=\cos A} and y = sin ⁡ A {\displaystyle y=\sin A} . This representation allows for

2277-440: Is to remember facts and relationships in trigonometry. For example, the sine , cosine , and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA: One way to remember the letters is to sound them out phonetically (i.e. / ˌ s oʊ k ə ˈ t oʊ ə / SOH -kə- TOH -ə , similar to Krakatoa ). Another method

2376-794: The Surya Siddhanta , and its properties were further documented in the 5th century (AD) by Indian mathematician and astronomer Aryabhata . These Greek and Indian works were translated and expanded by medieval Islamic mathematicians . In 830 AD, Persian mathematician Habash al-Hasib al-Marwazi produced the first table of cotangents. By the 10th century AD, in the work of Persian mathematician Abū al-Wafā' al-Būzjānī , all six trigonometric functions were used. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also made important innovations in spherical trigonometry The Persian polymath Nasir al-Din al-Tusi has been described as

2475-452: The 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top. This works because the distances from the 1 mark are proportional to the logarithms of the marked values. The illustration below demonstrates the computation of ⁠ 5.5 / 2 ⁠ . The 2 on the top scale is placed over the 5.5 on the bottom scale. The resulting quotient, 2.75 , can then be read below

2574-1162: The Fourier transform . This has applications to quantum mechanics and communications , among other fields. Trigonometry is useful in many physical sciences , including acoustics , and optics . In these areas, they are used to describe sound and light waves , and to solve boundary- and transmission-related problems. Other fields that use trigonometry or trigonometric functions include music theory , geodesy , audio synthesis , architecture , electronics , biology , medical imaging ( CT scans and ultrasound ), chemistry , number theory (and hence cryptology ), seismology , meteorology , oceanography , image compression , phonetics , economics , electrical engineering , mechanical engineering , civil engineering , computer graphics , cartography , crystallography and game development . Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. Identities involving only angles are known as trigonometric identities . Other equations, known as triangle identities , relate both

2673-746: The Global Positioning System and artificial intelligence for autonomous vehicles . In land surveying , trigonometry is used in the calculation of lengths, areas, and relative angles between objects. On a larger scale, trigonometry is used in geography to measure distances between landmarks. The sine and cosine functions are fundamental to the theory of periodic functions , such as those that describe sound and light waves. Fourier discovered that every continuous , periodic function could be described as an infinite sum of trigonometric functions. Even non-periodic functions can be represented as an integral of sines and cosines through

2772-503: The law of sines to Figure 3 produces Equation 1 . where Range plays no role in Equation 1 , which is true as long as the three assumptions are met. In fact, Equation 1 is the same equation solved by the mechanical sights of steerable torpedo tubes used on surface ships during World War I and World War II. Torpedo launches from steerable torpedo tubes meet the three stated assumptions well. However, an accurate torpedo launch from

2871-401: The law of tangents for spherical triangles, and provided proofs for both these laws. Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi . One of the earliest works on trigonometry by a northern European mathematician

2970-481: The law of the logarithm of a product , the product of the two numbers can be read. More elaborate slide rules can perform other calculations, such as square roots , exponentials , logarithms , and trigonometric functions . The user may estimate the location of the decimal point in the result by mentally interpolating between labeled graduations. Scientific notation is used to track the decimal point for more precise calculations. Addition and subtraction steps in

3069-555: The sinking of South Korea 's ROKS  Cheonan by North Korea in 2010, the last warship sunk by a submarine torpedo attack, ARA General Belgrano in 1982, was struck by two torpedoes from a three torpedo spread. To accurately compute the gyro angle for a torpedo in a general engagement scenario, the target course, speed, range, and bearing must be accurately known. During World War II, target course, range, and bearing estimates often had to be generated using periscope observations, which were highly subjective and error prone. The TDC

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3168-407: The trigonometric calculations required to compute a target intercept course for the torpedo. It also had an electromechanical interface to the torpedoes, allowing it to automatically set courses while torpedoes were still in their tubes, ready to be fired. The TDC's target tracking capability was used by the fire control party to continuously update the fire control solution even while the submarine

3267-707: The versine ( versin( θ ) = 1 − cos( θ ) = 2 sin ( ⁠ θ / 2 ⁠ ) ) (which appeared in the earliest tables ), the coversine ( coversin( θ ) = 1 − sin( θ ) = versin( ⁠ π / 2 ⁠ − θ ) ), the haversine ( haversin( θ ) = ⁠ 1 / 2 ⁠ versin( θ ) = sin ( ⁠ θ / 2 ⁠ ) ), the exsecant ( exsec( θ ) = sec( θ ) − 1 ), and the excosecant ( excsc( θ ) = exsec( ⁠ π / 2 ⁠ − θ ) = csc( θ ) − 1 ). See List of trigonometric identities for more relations between these functions. For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, predicting eclipses, and describing

3366-478: The 1930s for aircraft pilots to help with dead reckoning . With the aid of scales printed on the frame it also helps with such miscellaneous tasks as converting time, distance, speed, and temperature values, compass errors, and calculating fuel use. The so-called "prayer wheel" is still available in flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions,

3465-538: The 1950s and 1960s, even as desktop electronic computers were gradually introduced. But after the handheld scientific calculator was introduced in 1972 and became inexpensive in the mid-1970s, slide rules became largely obsolete , so most suppliers departed the business. In the United States , the slide rule is colloquially called a slipstick . Each ruler's scale has graduations labeled with precomputed outputs of various mathematical functions , acting as

3564-406: The 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where 7 is on the top scale: [REDACTED] But since the 7 is above the second set of numbers that number must be multiplied by 10 . Thus, even though the answer directly reads 1.4 , the correct answer is 1.4×10 = 14 . For an example with even larger numbers, to multiply 88×20 , the top scale

3663-512: The 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables ( Ptolemy's table of chords ) in Book 1, chapter 11 of his Almagest . Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. (The value we call sin(θ) can be found by looking up the chord length for twice

3762-419: The A scale). Slide the slide until the number on the D scale which is against 1 on the C cursor is the same as the number on the B cursor which is against the base number on the A scale. (Examples: A 8, B 2, C 1, D 2; A 27, B 3, C 1, D 3.) Quadratic equations of the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} can be solved by first reducing

3861-561: The British nuclear-powered submarine HMS  Conqueror — sank ARA  General Belgrano in 1982. During World War I , computing a target intercept course for a torpedo was a manual process where the fire control party was aided by various slide rules (the U.S. examples were the Mark VIII Angle Solver (colloquially called the "banjo", for its shape), and the "Is/Was" circular sliderule ( Nasmith Director ), for predicting where

3960-579: The CI scale is used. Common forms such as k sin ⁡ x {\displaystyle k\sin x} can be read directly from x on the S scale to the result on the D scale, when the C scale index is set at  k . For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/ radian . Inverse trigonometric functions are found by reversing

4059-466: The D scale and read its square on the A scale. Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale; to find the square root of nine, use the first one; the second one gives the square root of 90. For x y {\displaystyle x^{y}} problems, use

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4158-411: The D scale. The cursor is then moved along the rule until a position is found where the numbers on the CI and D scales add up to p {\displaystyle p} . These two values are the roots of the equation. The LLN scales can be used to compute and compare the cost or return on a fixed rate loan or investment. The simplest case is for continuously compounded interest. Example: Taking D as

4257-478: The E6B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of proficiency in its use. Proportion wheels are simple circular slide rules used in graphic design to calculate aspect ratios . Lining up the original and desired size values on the inner and outer wheels will display their ratio as a percentage in a small window. Though not as common since

4356-492: The LL scales. When several LL scales are present, use the one with x on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find y on the C scale and go down to the LL scale with x on it. That scale will indicate the answer. If y is "off the scale", locate x y / 2 {\displaystyle x^{y/2}} and square it using the A and B scales as described above. Alternatively, use

4455-480: The S scale with C (or D) scale. (On many closed-body rules the S scale relates to the A and B scales instead and covers angles from around 0.57 up to 90 degrees; what follows must be adjusted appropriately.) The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with the C (or D) scale for angles less than 45 degrees. For angles greater than 45 degrees

4554-526: The Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series . Also in the 18th century, Brook Taylor defined the general Taylor series . Trigonometric ratios are the ratios between edges of a right triangle. These ratios depend only on one acute angle of the right triangle, since any two right triangles with

4653-405: The TDC and other fire control equipment was mounted in the conning tower , which was a very small space. The packaging problem was severe and the performance of some early torpedo fire control equipment was hampered by the need to make it small. It had an array of handcranks, dials, and switches for data input and display. To generate a fire control solution, it required inputs on The TDC performed

4752-453: The TDC with Arma Corporation and Ford Instruments . This culminated in the "very complicated" Mark 1 in 1938. This was retrofitted into older boats, beginning with Dolphin and up through the newest Salmon s . The first submarine designed to use the TDC was Tambor , launched in 1940 with the Mark III, located in the conning tower . (This differed from earlier outfits.) It proved to be

4851-468: The Torpedo Data Computer manual. The Submarine Torpedo Fire Control Manual discusses the calculations in a general sense and a greatly abbreviated form of that discussion is presented here. The general torpedo fire control problem is illustrated in Figure 2. The problem is made more tractable if we assume: As can be seen in Figure 2, these assumptions are not true in general because of the torpedo ballistic characteristics and torpedo tube parallax. Providing

4950-452: The act of positioning the top scale to start at the bottom scale's label for x {\displaystyle x} corresponds to shifting the top logarithmic scale by a distance of log ⁡ ( x ) {\displaystyle \log(x)} . This aligns each top scale's number y {\displaystyle y} at offset log ⁡ ( y ) {\displaystyle \log(y)} with

5049-767: The advent of computerized layout, they are still made and used. In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the Breitling Navitimer . The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed , rate /time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer— nautical mile and gallon—liter fuel amount conversion functions. Trigonometry Trigonometry (from Ancient Greek τρίγωνον ( trígōnon )  'triangle' and μέτρον ( métron )  'measure')

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5148-509: The aim to simplify an expression, to find a more useful form of an expression, or to solve an equation . Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the Babylonians , studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used

5247-453: The angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine , Islamic , and, later, Western European worlds. The modern definition of the sine is first attested in

5346-592: The best torpedo fire control system of World War II . In 1943, the Torpedo Data Computer Mark IV was developed to support the Mark 18 torpedo. Both the Mk III and Mk IV TDC were developed by Arma Corporation (now American Bosch Arma). A straight-running torpedo has a gyroscope -based control system that ensures that the torpedo will run a straight course. The torpedo can run on a course different from that of

5445-564: The bottom scale's number at position log ⁡ ( x ) + log ⁡ ( y ) {\displaystyle \log(x)+\log(y)} . Because log ⁡ ( x ) + log ⁡ ( y ) = log ⁡ ( x × y ) {\displaystyle \log(x)+\log(y)=\log(x\times y)} , the mark on the bottom scale at that position corresponds to x × y {\displaystyle x\times y} . With x=2 and y=3 for example, by positioning

5544-415: The bow ." Angle on the bow is the angle formed by the target course and the line of sight to the submarine. Some skippers, like Richard O'Kane , practiced determining the angle on the bow by looking at Imperial Japanese Navy ship models mounted on a calibrated lazy Susan through an inverted binocular barrel. To generate target position data versus time, the TDC needed to solve the equations of motion for

5643-498: The calculation of commonly found trigonometric values, such as those in the following table: Using the unit circle , one can extend the definitions of trigonometric ratios to all positive and negative arguments (see trigonometric function ). The following table summarizes the properties of the graphs of the six main trigonometric functions: Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting

5742-552: The course deviations they cause are usually small enough to be ignorable. U.S. submarines during World War II preferred to fire their torpedoes at small gyro angles because the TDC's fire control solutions were most accurate for small angles. The problem of computing the gyro angle setting is a trigonometry problem that is simplified by first considering the calculation of the deflection angle, which ignores torpedo ballistics and parallax. For small gyro angles, θ Gyro ≈ θ Bearing − θ Deflection . A direct application of

5841-488: The creator of trigonometry as a mathematical discipline in its own right. He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure , he stated the law of sines for plane and spherical triangles, discovered

5940-448: The demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of

6039-498: The details as to how to correct the torpedo gyro angle calculation for ballistics and parallax is complicated and beyond the scope of this article. Most discussions of gyro angle determination take the simpler approach of using Figure 3, which is called the torpedo fire control triangle. Figure 3 provides an accurate model for computing the gyro angle when the gyro angle is small, usually less than 30°. The effects of parallax and ballistics are minimal for small gyro angle launches because

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6138-520: The device. Two upgraded World War II-era U.S. Navy fleet submarines ( USS  Tusk and Cutlass ) with their TDCs continue to serve with Taiwan's navy and U.S. Nautical Museum staff are assisting them with maintaining their equipment. The museum also has a fully restored and functioning TDC from USS  Pampanito , docked in San Francisco . The problem of aiming a torpedo has occupied military engineers since Robert Whitehead developed

6237-464: The difficulty in locating figures along a dish, and limited number of scales. Another drawback of circular slide rules is that less-important scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch. One slide rule remaining in daily use around the world is the E6B . This is a circular slide rule first created in

6336-474: The divisions mark a scale to a precision of two significant figures , and the user estimates the third figure. Some high-end slide rules have magnifier cursors that make the markings easier to see. Such cursors can effectively double the accuracy of readings, permitting a 10-inch slide rule to serve as well as a 20-inch model. Various other conveniences have been developed. Trigonometric scales are sometimes dual-labeled, in black and red, with complementary angles,

6435-423: The domain of a trigonometric function, however, they can be made invertible. The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table: When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series . For instance, sine and cosine have the following representations: With these definitions

6534-405: The earliest-known tables of values for trigonometric ratios (also called trigonometric functions ) such as sine . Throughout history, trigonometry has been applied in areas such as geodesy , surveying , celestial mechanics , and navigation . Trigonometry is known for its many identities . These trigonometric identities are commonly used for rewriting trigonometrical expressions with

6633-423: The equation to the form x 2 − p x + q = 0 {\displaystyle x^{2}-px+q=0} (where p = − b / a {\displaystyle p=-b/a} and q = c / a {\displaystyle q=c/a} ), and then aligning the index ("1") of the C scale to the value q {\displaystyle q} on

6732-520: The interest rate in percent, slide the index (the "1" at the right or left end of the scale) of C to the percent on D. The corresponding value on LL2 directly below the index will be the multiplier for 10 cycles of interest (typically years). The value on LL2 below 2 on the C scale will be the multiplier after 20 cycles, and so on. The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees. For angles from around 5.7 up to 90 degrees, sines are found by comparing

6831-647: The log of a value on a multiplier scale), natural logarithm (ln) and exponential ( e ) scales. Others feature scales for calculating hyperbolic functions . On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order. The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions. There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute x 2 {\displaystyle x^{2}} , for example, locate x on

6930-519: The modern torpedo in the 1860s. These early torpedoes ran at a preset depth on a straight course (consequently they are frequently referred to as "straight runners"). This was the state of the art in torpedo guidance until the development of the homing torpedo during the latter part of World War II . The vast majority of submarine torpedoes during World War II were straight running, and these continued in use for many years after World War II. In fact, two World War II-era straight running torpedoes — fired by

7029-421: The orbits of the planets. In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, as well as in satellite navigation systems . Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. Trigonometry is still used in navigation through such means as

7128-612: The other as its actual operator. Despite these drawbacks, the use of the TDC was an important factor in the successful commerce raiding program conducted by American submarines during the Pacific campaign of World War II. Accounts of the American submarine campaign in the Pacific often cite the use of TDC. Some officers became highly skilled in its use, and the Navy set up a training school for operation of

7227-593: The other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 6, 4.5 is over 6, and 6 is over 8, among other pairs. For a real-life situation where 750 represents a whole 100%, these readings could be interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%. In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular are trigonometric , usually sine and tangent , common logarithm (log 10 ) (for taking

7326-458: The outer rings. Rather than "split" scales, high-end circular rules use spiral scales for more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale. Around 1970, an inexpensive model from B. C. Boykin (Model 510) featured 20 scales, including 50-inch C-D (multiplication) and log scales. The RotaRule featured a friction brake for the cursor. The main disadvantages of circular slide rules are

7425-442: The point of minimum deflection angle sensitivity to track angle errors for a given target speed. This minimum occurs at the points of zero slope on the curves in Figure 4 (these points are marked by small triangles). The curves show the solutions of Equation 2 for deflection angle as a function of target speed and track angle. Figure 4 confirms that 110° is the optimum track angle for a 16-knot (30 km/h) target, which would be

7524-404: The process. Many slide rules have S, T, and ST scales marked with degrees and minutes (e.g. some Keuffel and Esser models (Doric duplex 5" models, for example), late-model Teledyne-Post Mannheim-type rules). So-called decitrig models use decimal fractions of degrees instead. Base-10 logarithms and exponentials are found using the L scale, which is linear. Some slide rules have a Ln scale, which

7623-888: The range 1 to 10 is a single decade, and the range from 10 to 100 is another decade. Thus, single-decade scales (named C and D) range from 1 to 10 across the entire length of the slide rule, while double-decade scales (named A and B) range from 1 to 100 over the length of the slide rule. The following logarithmic identities transform the operations of multiplication and division to addition and subtraction, respectively: log ⁡ ( x × y ) = log ⁡ ( x ) + log ⁡ ( y ) , {\displaystyle \log(x\times y)=\log(x)+\log(y)\,,} log ⁡ ( x / y ) = log ⁡ ( x ) − log ⁡ ( y ) . {\displaystyle \log(x/y)=\log(x)-\log(y)\,.} With two logarithmic scales,

7722-456: The result, 8.25 , can be read beneath the 3 in the top scale in the figure above, without the need to register the intermediate result for ⁠ 5.5 / 2 ⁠ . Because pairs of numbers that are aligned on the logarithmic scales form constant ratios, no matter how the scales are offset, slide rules can be used to generate equivalent fractions that solve proportion and percent problems. For example, setting 7.5 on one scale over 10 on

7821-466: The rightmost 1 on the C scale, and read the answer off the next higher LL scale. For example, aligning the rightmost 1 on the C scale with 2 on the LL2 scale, 3 on the C scale lines up with 8 on the LL3 scale. To extract a cube root using a slide rule with only C/D and A/B scales, align 1 on the B cursor with the base number on the A scale (taking care as always to distinguish between the lower and upper halves of

7920-400: The same acute angle are similar . So, these ratios define functions of this angle that are called trigonometric functions . Explicitly, they are defined below as functions of the known angle A , where a , b and h refer to the lengths of the sides in the accompanying figure: The hypotenuse is the side opposite to the 90-degree angle in a right triangle; it is the longest side of

8019-453: The scale width is narrowed to make room for end margins. Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm (0.0039 in) off-centre of the pivot can result in a 0.2 mm (0.0079 in) worst case alignment error. The pivot does prevent scratching of the face and cursors. The highest accuracy scales are placed on

8118-391: The sides and angles of a given triangle. In the following identities, A , B and C are the angles of a triangle and a , b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram). The law of sines (also known as the "sine rule") for an arbitrary triangle states: where Δ {\displaystyle \Delta }

8217-430: The sine, tangent, and secant of the complementary angle abbreviated to "co-". With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines . These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. A common use of mnemonics

8316-413: The slide rule in the 17th century based on the emerging work on logarithms by John Napier . It made calculations faster and less error-prone than evaluating on paper . Before the advent of the scientific pocket calculator , it was the most commonly used calculation tool in science and engineering . The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through

8415-440: The slide rule is quoted in terms of the nominal width of the scales. Scales on the most common "10-inch" models are actually 25 cm, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflows. Pocket rules are typically 5 inches (12 cm). Models a couple of metres (yards) wide were made to be hung in classrooms for teaching purposes. Typically

8514-582: The so-called "Darmstadt" style. Duplex slide rules often duplicate some of the scales on the back. Scales are often "split" to get higher accuracy. For example, instead of reading from an A scale to a D scale to find a square root, it may be possible to read from a D scale to an R1 scale running from 1 to square root of 10 or to an R2 scale running from square root of 10 to 10, where having more subdivisions marked can result in being able to read an answer with one more significant digit. Circular slide rules come in two basic types, one with two cursors, and another with

8613-402: The submarine by adjusting a parameter called the gyro angle, which sets the course of the torpedo relative to the course of the submarine (see Figure 2). The primary role of the TDC is to determine the gyro angle setting required to ensure that the torpedo will strike the target. Determining the gyro angle required the real-time solution of a complex trigonometric equation (see Equation 1 for

8712-399: The submarine to launch the torpedo on a course different from that of the submarine, which is important tactically. Otherwise, the submarine would need to be pointed at the projected intercept point in order to launch a torpedo. Requiring the entire vessel to be pointed in order to launch a torpedo would be time consuming, require precise submarine course control, and would needlessly complicate

8811-409: The target bearing angle is different from the point of view of the periscope versus the point of view of the torpedo, which is referred to as torpedo tube parallax. These factors are a significant complication in the calculation of the gyro angle, and the TDC must compensate for their effects. Straight running torpedoes were usually launched in salvo (i.e. multiple launches in a short period of time) or

8910-415: The target relative to the submarine. The equations of motion are differential equations and the TDC used mechanical integrators to generate its solution. The TDC needed to be positioned near other fire control equipment to minimize the amount of electromechanical interconnect. Because submarine space within the pressure hull was limited, the TDC needed to be as small as possible. On World War II submarines,

9009-405: The top scale to start at the bottom scale's 2 , the result of the multiplication 3×2=6 can then be read on the bottom scale under the top scale's 3 : [REDACTED] While the above example lies within one decade, users must mentally account for additional zeroes when dealing with multiple decades. For example, the answer to 7×2=14 is found by first positioning the top scale to start above

9108-534: The top scale's 1 : [REDACTED] There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end. With more complex calculations involving multiple factors in the numerator and denominator of an expression, movement of the scales can be minimized by alternating divisions and multiplications. Thus ⁠ 5.5×3 / 2 ⁠ would be computed as ⁠ 5.5 / 2 ⁠ ×3 and

9207-524: The torpedo firing process. The TDC with target tracking gives the submarine the ability to maneuver independently of the required target intercept course for the torpedo. As is shown in Figure 2, in general, the torpedo does not actually move in a straight path immediately after launch and it does not instantly accelerate to full speed, which are referred to as torpedo ballistic characteristics. The ballistic characteristics are described by three parameters: reach, turning radius, and corrected torpedo speed. Also,

9306-528: The triangle and one of the two sides adjacent to angle A . The adjacent leg is the other side that is adjacent to angle A . The opposite side is the side that is opposite to angle A . The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under Mnemonics . The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively: The cosine, cotangent, and cosecant are so named because they are respectively

9405-506: The trigonometric functions can be defined for complex numbers . When extended as functions of real or complex variables, the following formula holds for the complex exponential: This complex exponential function, written in terms of trigonometric functions, is particularly useful. Trigonometric functions were among the earliest uses for mathematical tables . Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between

9504-440: The trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions. In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord ( crd( θ ) = 2 sin( ⁠ θ / 2 ⁠ ) ),

9603-426: The values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). Most allow a choice of angle measurement methods: degrees , radians, and sometimes gradians . Most computer programming languages provide function libraries that include

9702-427: Was maneuvering. The TDC's target tracking ability also allowed the submarine to accurately fire torpedoes even when the target was temporarily obscured by smoke or fog. Since the TDC actually performed two separate functions, generating target position estimates and computing torpedo firing angles, the TDC actually consisted of two types of analog computers: The equations implemented in the angle solver can be found in

9801-428: Was used to refine the estimates of the target's course, range, and bearing through a process of Estimating the target's course was generally considered the most difficult of the observation tasks. The accuracy of the result was highly dependent on the experience of the skipper. During combat, the actual course of the target was not usually determined but instead the skippers determined a related quantity called " angle on

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