Saturday, June 28, 2014

CONDITIONAL SENTENCES

CONDITIONAL    SENTENCES.

INTRODUCTION:
For the non-custodial punishment for a crime in Canada, see Conditional sentence (Canada).
Conditional sentences are sentences expressing factual implications, or hypothetical situations and their consequences. They are so called because the validity of the main clause of the sentence is conditional on the existence of certain circumstances, which may be expressed in a dependent clause or may be understood from the context.
A full conditional sentence (one which expresses the condition as well as its consequences) therefore contains two clauses: the dependent clause expressing the condition, called the protasis; and the main clause expressing the consequence, called the apodosis.[1] An example of such a sentence (in English) is the following:
If it rains the picnic will be cancelled.
Here the condition is expressed by the clause "If it rains", this being the protasis, while the consequence is expressed by "the picnic will be cancelled", this being the apodosis. (The protasis may either precede or follow the apodosis; it is equally possible to say "The picnic will be cancelled if it rains".) In terms of logic, the protasis corresponds to the antecedent, and the apodosis to the consequent.
Languages use a variety of grammatical forms and constructions in conditional sentences. The forms of verbs used in the protasis and apodosis are often subject to particular rules as regards their tense and mood. Many languages have a specialized type of verb form called the conditional mood – broadly equivalent in meaning to the English "would (do something)" – for use in some types of conditional sentence.

Types of conditional sentence

There are various ways of classifying conditional sentences. One distinction is between those that state an implication between facts, and those that set up and refer to a hypothetical situation. There is also the distinction between conditionals that are considered factual or predictive, and those that are considered counterfactual or speculative (referring to a situation that did not or does not really exist).

Implicative and predictive

A conditional sentence expressing an implication (also called a factual conditional sentence) essentially states that if one fact holds, then so does another. (If the sentence is not a declarative sentence, then the consequence may be expressed as an order or a question rather than a statement.) The facts are usually stated in whatever grammatical tense is appropriate to them; there are not normally special tense or mood patterns for this type of conditional sentence. Such sentences may be used to express a certainty, a universal statement, a law of science, etc. (in these cases if may often be replaced by when):
If you heat water to 100 degrees, it boils.
If the sea is stormy, the waves are high.
They can also be used for logical deductions about particular circumstances (which can be in various mixtures of past, present and future):
If it's raining here now, then it was raining on the West Coast this morning.
If it's raining now, then your laundry is getting wet.
If it's raining now, there will be mushrooms to be picked next week.
If he locked the door, then Kitty is trapped inside.
A predictive conditional sentence concerns a situation dependent on a hypothetical (but entirely possible) future event. The consequence is normally also a statement about the future, although it may also be a consequent statement about present or past time (or a question or order).
If I become President, I'll lower taxes.
If it rains this afternoon, everybody will stay home.
If it rains this afternoon, then yesterday's weather forecast was wrong.
If it rains this afternoon, your garden party is doomed.
What will you do if he invites you?
If you see them, shoot!

Counterfactual

Main article: Counterfactual conditional
In a counterfactual or speculative[2] conditional sentence, a situation is described as dependent on a condition that is known to be false, or presented as unlikely. The time frame of the hypothetical situation may be past, present or future, and the time frame of the condition does not always correspond to that of the consequence. For example:
If I were king, I could have you thrown in the dungeon.
If I won the lottery, I would buy a car.
If he said that to me, I would run away.
If you had called me, I would have come.
If you had done your job properly, we wouldn't be in this mess now.
The difference in meaning between a "counterfactual" conditional with a future time frame, and a "predictive" conditional as described in the previous section, may be slight. For example, there is no great practical difference in meaning between "If it rained tomorrow, I would cancel the match" and "If it rains tomorrow, I will cancel the match".
It is in the counterfactual type of conditional sentence that the grammatical form called the conditional mood (meaning something like the English "would ...") is most often found. For the uses of particular verb forms and grammatical structures in the various types and parts of conditional sentences in certain languages, see the following sections.

Grammar of conditional sentences

Languages have different rules concerning the grammatical structure of conditional sentences. These may concern the syntactic structure of the condition clause (protasis) and consequence (apodosis), as well as the forms of verbs used in them (particularly their tense and mood). Rules for English and certain other languages are described below; more information can be found in the articles on the grammars of individual languages. (Some languages are also described in the article on the conditional mood.)

English

Main article: English conditional sentences
In English conditional sentences, the condition clause (protasis) is most commonly introduced by the conjunction if, or sometimes other conjunctions or expressions such as unless, provided (that), providing (that) and as long as. Certain condition clauses can also be formulated using inversion without any conjunction (should you fail...; were he to die...; had they helped us...).
In English language teaching, conditional sentences are often classified under the headings zero conditional, first conditional (or conditional I), second conditional (or conditional II), third conditional (or conditional III) and mixed conditional, according to the grammatical pattern followed.[3] A range of variations on these structures are possible.

Zero conditional

"Zero conditional" refers to conditional sentences that express a simple implication (see above section), particularly when both clauses are in the present tense:
If you don't eat for a long time, you become hungry.
This form of the conditional expresses the idea that a universally known fact is being described:
If you touch a flame, you burn yourself.
The act of burning oneself only happens on the condition of the first clause being completed. However such sentences can be formulated with a variety of tenses (and moods), as appropriate to the situation.

First conditional

"First conditional" refers to predictive conditional sentences (see above section); here, normally, the condition is expressed using the present tense and the consequence using the future:
If you make a mistake, someone will let you know.

Second conditional

"Second conditional" refers to the pattern where the condition clause is in the past tense, and the consequence in conditional mood (using would or, in the first person and rarely, should). This is used for hypothetical, counterfactual situations in a present or future time frame (where the condition expressed is known to be false or is presented as unlikely).
If I liked parties, I would attend more of them.
If it were to rain tomorrow, I would dance in the street.
The past tense used in the condition clause is historically the past subjunctive; however in modern English this is identical to the past indicative except in certain dialects in the case of the verb be (first and third person singular), where the indicative is was and the subjunctive were. In this case either form may be used (was is more colloquial, and were more formal, although the phrase if I were you is common in colloquial language too):
If I (he, she, it) was/were rich, there would be plenty of money available for this project.

Third conditional

"Third conditional" is the pattern where the condition clause is in the past perfect, and the consequence is expressed using the conditional perfect. This is used to refer to hypothetical, counterfactual (or believed likely to be counterfactual) situations in the past
If you had called me, I would have come.

Mixed conditionals

"Mixed conditional" usually refers to a mixture of the second and third conditionals (the counterfactual patterns). Here either the condition or the consequence, but not both, has a past time reference:
If you had done your job properly, we wouldn't be in this mess now.
If we were soldiers, we wouldn't have done it like that.

Latin

Conditional sentences in Latin are traditionally classified into three categories, based on grammatical structure.
  • simple conditions (factual or logical implications)
    • present tense [if present indicative then indicative]
    • past tense [if perfect/imperfect indicative then indicative]
  • future conditions
    • "future more vivid" [if future indicative then future indicative]
    • "future less vivid" [if present subjunctive then present subjunctive]
  • contrafactual conditions
    • "present contrary-to-fact" [if imperfect subjunctive then imperfect subjunctive]
    • "past contrary-to-fact" [if pluperfect subjunctive then pluperfect subjunctive]

French

In French, the conjunction corresponding to "if" is si. The use of tenses is quite similar to English:
  • In implicative conditional sentences, the present tense (or other appropriate tense, mood, etc.) is used in both clauses.
  • In predictive conditional sentences, the future tense or imperative generally appears in the main clause, but the condition clause is formed with the present tense (as in English). This contrasts with dependent clauses introduced by certain other conjunctions, such as quand ("when"), where French uses the future (while English has the present).
  • In counterfactual conditional sentences, the imperfect is used to express the condition (where English similarly uses the past tense). The main clause contains the conditional mood (e.g. j'arriverais, "I would arrive").
  • In counterfactual conditional sentences with a past time frame, the condition is expressed using the pluperfect e.g. (s'il avait attendu, "if he had waited"), and the consequence with the conditional perfect (e.g. je l'aurais vu, "I would have seen him"). Again these verb forms parallel those used in English.
As in English, certain mixtures and variations of these patterns are possible. See also French verbs.

Italian

Italian uses the following patterns (the equivalent of "if" is se):
  • Present tense (or other as appropriate) in both parts of an implicative conditional.
  • Future tense in both parts of a predictive conditional sentence (the future is not replaced with the present in condition clauses as in English or French).
  • In a counterfactual conditional, the imperfect subjunctive is used for the condition, and the conditional mood for the main clause. A more informal equivalent is to use the imperfect indicative in both parts.
  • In a counterfactual conditional with past time frame, the pluperfect subjunctive is used for the condition, and the past conditional (conditional perfect) for the main clause.
See also Italian verbs.

Slavic languages

In Slavic languages, such as Russian, clauses in conditional sentences generally appear in their natural tense (future tense for future reference, etc.) However, for counterfactuals, a conditional/subjunctive marker such as the Russian бы by generally appears in both condition and consequent clauses, and this normally accompanies the past tense form of the verb.
See Russian grammar, Bulgarian grammar, etc. for more detail.

Logic

While the material conditional operator used in logic (i.e.\scriptstyle p \Rightarrow q) is sometimes read aloud in the form of a conditional sentence (i.e. "if p, then q"), the intuitive interpretation of conditional statements in natural language does not always correspond to the definition of this mathematical relation. Modelling the meaning of real conditional statements requires the definition of an indicative conditional, and contrary-to-fact statements require a counterfactual conditional operator, formalized in modal logic.

See also

References

  1. Haspelmath, Martin; König, Ekkehard; Oesterreicher, Wulf; Raible, Wolfgang: Language Typology and Language Universals, Walter de Gruyter, 2001, p. 1002.
  2. Mead, Hayden; Stevenson, Jay (1996), The Essentials of Grammar, New York: Berkley Books, p. 55, ISBN 978-0-425-15446-5, OCLC 35301673
  3. Craig Thane, Teacher Training Essentials: Workshops for Professional Development, Cambridge University Press, 2010, p. 67.

External links

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TRIGONOMETRY.

TRIGONOMETRY.

INTRODUCTION:
"Trig" redirects here. For other uses, see Trig (disambiguation).
The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles.
Trigonometry
Reference
Laws and theorems
Calculus
Trigonometry (from Greek trigōnon, "triangle" + metron, "measure"[1]) is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies.[2]
The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of the practical art of surveying.
Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry (a fundamental part of astronomy and navigation). Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.
Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course.

History

Main article: History of trigonometry
Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".[3]
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.[4] 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 a similar method.[5] The ancient Greeks transformed trigonometry into an ordered science.[6]
In the 3rd century BCE, classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.[7]
The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century (CE) Indian mathematician and astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th-century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.[10] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[11] 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 James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylor series.[13]

Overview

Main article: Trigonometric function
In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
  • Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.
  • Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.
  • Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of 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. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:
\csc A=\frac{1}{\sin A}=\frac{c}{a} ,
\sec A=\frac{1}{\cos A}=\frac{c}{b} ,
\cot A=\frac{1}{\tan A}=\frac{\cos A}{\sin A}=\frac{b}{a} .
The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively 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. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

Extending the definitions

Fig. 1a – Sine and cosine of an angle θ defined using the unit circle.
The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.
e^{x+iy} = e^x(\cos y + i \sin y).
See Euler's and De Moivre's formulas.
  • Graphing process of y = sin(x) using a unit circle.
  • Graphing process of y = csc(x), the reciprocal of sine, using a unit circle.
  • Graphing process of y = tan(x) using a unit circle.

Mnemonics

Main article: Mnemonics in trigonometry
A common use of mnemonics 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:[14]
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-kə-toe-uh' /soʊkəˈtoʊə/). Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".[15]

Calculating trigonometric functions

Main article: Generating trigonometric tables
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 the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today 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 grad. Most computer programming languages provide function libraries that include 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.[16]

Applications of trigonometry

Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
Main article: Uses of trigonometry
There is an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthesis, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, image compression, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

Pythagorean identities

Identities are those equations that hold true for any value.
\sin^2 A + \cos^2 A = 1 \
(Note that the following two can be derived from the first)
\sec^2 A - \tan^2 A = 1 \
\csc^2 A - \cot^2 A = 1 \

Angle transformation formulas

\sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B
\cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B
\tan (A \pm B) = \frac{ \tan A \pm \tan B }{ 1 \mp \tan A \ \tan B}
\cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A }

Common formulas

Triangle with sides a,b,c and respectively opposite angles A,B,C
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.
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.

Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states:
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,
where R is the radius of the circumscribed circle of the triangle:
R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.
Another law involving sines can be used to calculate the area of a triangle. Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:
\mbox{Area} = \frac{1}{2}a b\sin C.
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
c^2=a^2+b^2-2ab\cos C ,\,
or equivalently:
\cos C=\frac{a^2+b^2-c^2}{2ab}.\,
The law of cosines may be used to prove Heron's Area Formula, which is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
s=\frac{1}{2}(a+b+c),
then the area of the triangle is:
\mbox{Area} = \sqrt{s(s-a)(s-b)(s-c)}.

Law of tangents

The law of tangents:
\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}

Euler's formula

Euler's formula, which states that e^{ix} = \cos x + i \sin x, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
\sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.

See also

References

  1. "trigonometry". Online Etymology Dictionary.
  2. R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002)
  3. Boyer (1991). "Greek Trigonometry and Mensuration". p. 162.
  4. Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-387-95136-9
  5. Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag. pp. 744–. ISBN 978-3-540-06995-9.
  6. "The Beginnings of Trigonometry". Rutgers, The State University of New Jersey.
  7. Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004). Sherlock Holmes in Babylon: and other tales of mathematical history. MAA. p. 36. ISBN 0-88385-546-1
  8. Boyer p. 215
  9. Boyer pp. 237, 274
  10. Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 0-393-32030-8.
  11. Robert E. Krebs (2004). Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissnce. Greenwood Publishing Group. pp. 153–. ISBN 978-0-313-32433-8.
  12. William Bragg Ewald (2008). From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University Press US. p. 93. ISBN 0-19-850535-3
  13. Kelly Dempski (2002). Focus on Curves and Surfaces. p. 29. ISBN 1-59200-007-X
  14. Weisstein, Eric W., "SOHCAHTOA", MathWorld.
  15. A sentence more appropriate for high schools is "Some old horse came a'hopping through our alley". Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN 0-19-280675-0.
  16. Intel® 64 and IA-32 Architectures Software Developer’s Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C. Intel. 2013.

Bibliography

External links

Find more about Trigonometry at Wikipedia's sister projects
Definitions and translations from Wiktionary
Media from Commons
Quotations from Wikiquote
Source texts from Wikisource
Textbooks from Wikibooks
Learning resources from Wikiversity
Library resources about
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