tag:blogger.com,1999:blog-69123191961365013022024-03-08T08:19:53.565-08:00Eny ChaiiankEnyhttp://www.blogger.com/profile/14412948294694250246noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-6912319196136501302.post-70923487183441943872010-01-15T19:43:00.000-08:002010-01-15T20:13:23.185-08:00<b>rigonometry</b> (from <a href="http://en.wikipedia.org/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> <i><a href="http://en.wiktionary.org/wiki/%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF%CE%BD" class="extiw" title="wikt:τρίγωνον">trigōnon</a></i> "triangle" + <i><a href="http://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%84%CF%81%CE%BF%CE%BD" class="extiw" title="wikt:μέτρον">metron</a></i> "measure")<sup id="cite_ref-0" class="reference"><a href="http://en.wikipedia.org/wiki/Trigonometry#cite_note-0"><span>[</span>1<span>]</span></a></sup> is a branch of <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a> that studies <a href="http://en.wikipedia.org/wiki/Triangle_%28geometry%29" title="Triangle (geometry)" class="mw-redirect">triangles</a>, particularly <a href="http://en.wikipedia.org/wiki/Right_triangle" title="Right triangle">right triangles</a>. Trigonometry deals with relationships between the sides and the angles of triangles and with the <a href="http://en.wikipedia.org/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, which describe those relationships, as well as describing angles in general and the motion of <a href="http://en.wikipedia.org/wiki/Wave" title="Wave">waves</a> such as sound and light waves. <p>Trigonometry is usually taught in <a href="http://en.wikipedia.org/wiki/Secondary_schools" title="Secondary schools" class="mw-redirect">secondary schools</a> either as a separate course or as part of a <a href="http://en.wikipedia.org/wiki/Precalculus" title="Precalculus">precalculus</a> course. It has applications in both <a href="http://en.wikipedia.org/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a> and in <a href="http://en.wikipedia.org/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>, where it is essential in many branches of science and technology. A branch of trigonometry, called <a href="http://en.wikipedia.org/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a>, studies triangles on <a href="http://en.wikipedia.org/wiki/Sphere" title="Sphere">spheres</a>, and is important in <a href="http://en.wikipedia.org/wiki/Astronomy" title="Astronomy">astronomy</a> and <a href="http://en.wikipedia.org/wiki/Navigation" title="Navigation">navigation</a>.</p> <table id="toc" class="toc"> <tbody><tr> <td><br /></td></tr></tbody></table><br /><div class="thumb tright"> <div class="thumbinner" style="width: 242px;"><a href="http://en.wikipedia.org/wiki/File:TrigonometryTriangle.svg" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/TrigonometryTriangle.svg/240px-TrigonometryTriangle.svg.png" class="thumbimage" width="240" height="180" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:TrigonometryTriangle.svg" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" alt="" width="15" height="11" /></a></div> In this right triangle: <span style="white-space: nowrap;">sin <i>A</i> = <i>a</i>/<i>c</i>;</span> <span style="white-space: nowrap;">cos <i>A</i> = <i>b</i>/<i>c</i>;</span> <span style="white-space: nowrap;">tan <i>A</i> = <i>a</i>/<i>b</i>.</span></div> </div> </div> <p>If one <a href="http://en.wikipedia.org/wiki/Angle" title="Angle">angle</a> 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 <a href="http://en.wikipedia.org/wiki/Complementary_angles" title="Complementary angles">complementary angles</a>. The <a href="http://en.wikipedia.org/wiki/Shape" title="Shape">shape</a> of a right triangle is completely determined, up to <a href="http://en.wikipedia.org/wiki/Similarity_%28geometry%29" title="Similarity (geometry)">similarity</a>, by the angles. This means that once one of the other angles is known, the <a href="http://en.wikipedia.org/wiki/Ratio" title="Ratio">ratios</a> of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following <a href="http://en.wikipedia.org/wiki/Trigonometric_function" title="Trigonometric function" class="mw-redirect">trigonometric functions</a> of the known angle <i>A</i>, where <i>a</i>, <i>b</i> and <i>c</i> refer to the lengths of the sides in the accompanying figure:</p> <ul><li>The <b>sine</b> function (sin), defined as the ratio of the side opposite the angle to the <a href="http://en.wikipedia.org/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a>.</li></ul> <dl><dd> <dl><dd><img class="tex" alt="\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,." src="http://upload.wikimedia.org/math/7/a/3/7a397c5e3df9a0b03724f31dd5bd0d8b.png" /></dd></dl> </dd></dl> <ul><li>The <b>cosine</b> function (cos), defined as the ratio of the adjacent leg to the hypotenuse.</li></ul> <dl><dd> <dl><dd><img class="tex" alt="\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,." src="http://upload.wikimedia.org/math/5/5/b/55b70a8807539373af0b1f17705ea229.png" /></dd></dl> </dd></dl> <ul><li>The <b>tangent</b> function (tan), defined as the ratio of the opposite leg to the adjacent leg.</li></ul> <dl><dd> <dl><dd><img class="tex" alt="\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,." src="http://upload.wikimedia.org/math/f/6/2/f6297d65b32ed8e48a15c902bd41cbaa.png" /></dd></dl> </dd></dl> <p>The <b>hypotenuse</b> 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 <i>A</i>. The <b>adjacent leg</b> is the other side that is adjacent to angle <i>A</i>. The <b>opposite side</b> is the side that is opposite to angle <i>A</i>. The terms <b>perpendicular</b> and <b>base</b> 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 <a href="http://en.wikipedia.org/wiki/Trigonometry#Mnemonics">Mnemonics</a>).</p> <p>The <a href="http://en.wikipedia.org/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> of these functions are named the <b>cosecant</b> (csc or cosec), <b>secant</b> (sec) and <b>cotangent</b> (cot), respectively. The <a href="http://en.wikipedia.org/wiki/Inverse_trigonometric_function" title="Inverse trigonometric function" class="mw-redirect">inverse functions</a> are called the <b>arcsine</b>, <b>arccosine</b>, and <b>arctangent</b>, respectively. There are arithmetic relations between these functions, which are known as <a href="http://en.wikipedia.org/wiki/Trigonometric_identities" title="Trigonometric identities" class="mw-redirect">trigonometric identities</a>.</p> <p>With these functions one can answer virtually all questions about arbitrary triangles by using the <a href="http://en.wikipedia.org/wiki/Law_of_sines" title="Law of sines">law of sines</a> and the <a href="http://en.wikipedia.org/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a>. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every <a href="http://en.wikipedia.org/wiki/Polygon" title="Polygon">polygon</a> may be described as a finite combination of triangles.</p> <table class="gallery" cellpadding="0" cellspacing="0"> <tbody><tr> <td> <div class="gallerybox" style="width: 155px;"> <div class="thumb" style="padding: 52px 0pt; width: 150px;"> <div style="margin-left: auto; margin-right: auto; width: 120px;"><a href="http://en.wikipedia.org/wiki/File:Sin_drawing_process.gif" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/7/7d/Sin_drawing_process.gif" width="120" height="42" /></a></div> </div> <div class="gallerytext"> <p>Graphing process of <i>y</i> = sin(<i>x</i>) using a unit circle.</p> </div> </div> </td> <td> <div class="gallerybox" style="width: 155px;"> <div class="thumb" style="padding: 18px 0pt; width: 150px;"> <div style="margin-left: auto; margin-right: auto; width: 120px;"><a href="http://en.wikipedia.org/wiki/File:Tan_drawing_process.gif" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/e/ee/Tan_drawing_process.gif" width="120" height="109" /></a></div> </div> <div class="gallerytext"> <p>Graphing process of <i>y</i> = tan(<i>x</i>) using a unit circle.</p> </div> </div> </td> <td> <div class="gallerybox" style="width: 155px;"> <div class="thumb" style="padding: 18px 0pt; width: 150px;"> <div style="margin-left: auto; margin-right: auto; width: 120px;"><a href="http://en.wikipedia.org/wiki/File:Csc_drawing_process.gif" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/5/5d/Csc_drawing_process.gif" width="120" height="109" /></a></div> </div> <div class="gallerytext"> <p>Graphing process of <i>y</i> = csc(<i>x</i>) using a unit circle.</p> </div> </div> </td> </tr> </tbody></table> <h3><span class="editsection"></span><span class="mw-headline" id="Extending_the_definitions">Extending the definitions</span></h3> <div class="thumb tright"> <div class="thumbinner" style="width: 302px;"><a href="http://en.wikipedia.org/wiki/File:Sine_cosine_plot.svg" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/38/Sine_cosine_plot.svg/300px-Sine_cosine_plot.svg.png" class="thumbimage" width="300" height="200" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Sine_cosine_plot.svg" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" alt="" width="15" height="11" /></a></div> Graphs of the functions sin(<i>x</i>) and cos(<i>x</i>), where the angle <i>x</i> is measured in radians.</div> </div> </div> <p>The above definitions apply to angles between 0 and 90 degrees (0 and π/2 <a href="http://en.wikipedia.org/wiki/Radian" title="Radian">radians</a>) only. Using the <a href="http://en.wikipedia.org/wiki/Unit_circle" title="Unit circle">unit circle</a>, one can extend them to all positive and negative arguments (see <a href="http://en.wikipedia.org/wiki/Trigonometric_function" title="Trigonometric function" class="mw-redirect">trigonometric function</a>). The trigonometric functions are <a href="http://en.wikipedia.org/wiki/Periodic_function" title="Periodic function">periodic</a>, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals.</p> <p>The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a> and <a href="http://en.wikipedia.org/wiki/Infinite_series" title="Infinite series" class="mw-redirect">infinite series</a>. With these definitions the trigonometric functions can be defined for <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex numbers</a>. The complex function <b>cis</b> is particularly useful</p> <dl><dd><img class="tex" alt="\operatorname{cis}\,x = \cos x + i\sin x \! = e^{ix}. " src="http://upload.wikimedia.org/math/f/3/a/f3aecc112d5f1be37750697f9718b2d7.png" /></dd></dl> <p>See <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula">Euler's</a> and <a href="http://en.wikipedia.org/wiki/De_Moivre%27s_formula" title="De Moivre's formula">De Moivre's</a> formulas.</p> <h3><span class="editsection"></span><span class="mw-headline" id="Mnemonics">Mnemonics</span></h3> <p>A common use of <a href="http://en.wikipedia.org/wiki/Mnemonic" title="Mnemonic">mnemonics</a> is to remember facts and relationships in trigonometry. For example, the <i>sine</i>, <i>cosine</i>, and <i>tangent</i> ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA.</p> <dl><dd><b>S</b>ine = <b>O</b>pposite ÷ <b>H</b>ypotenuse</dd><dd><b>C</b>osine = <b>A</b>djacent ÷ <b>H</b>ypotenuse</dd><dd><b>T</b>angent = <b>O</b>pposite ÷ <b>A</b>djacent</dd></dl> <p>The memorization of this mnemonic can be aided by expanding it into a phrase, such as "<b>S</b>ome <b>O</b>fficers <b>H</b>ave <b>C</b>urly <b>A</b>uburn <b>H</b>air <b>T</b>ill <b>O</b>ld <b>A</b>ge".<sup id="cite_ref-3" class="reference"><a href="http://en.wikipedia.org/wiki/Trigonometry#cite_note-3"><span>[</span>4<span>]</span></a></sup> Any memorable phrase constructed of words beginning with the letters S-O-H-C-A-H-T-O-A will serve.</p> <h3><span class="editsection"></span><span class="mw-headline" id="Calculating_trigonometric_functions">Calculating trigonometric functions</span></h3> <div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Generating_trigonometric_tables" title="Generating trigonometric tables">Generating trigonometric tables</a></div> <p>Trigonometric functions were among the earliest uses for <a href="http://en.wikipedia.org/wiki/Mathematical_table" title="Mathematical table">mathematical tables</a>. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to <a href="http://en.wikipedia.org/wiki/Interpolate" title="Interpolate" class="mw-redirect">interpolate</a> between the values listed to get higher accuracy. <a href="http://en.wikipedia.org/wiki/Slide_rule" title="Slide rule">Slide rules</a> had special scales for trigonometric functions.</p> <p>Today <a href="http://en.wikipedia.org/wiki/Scientific_calculator" title="Scientific calculator">scientific calculators</a> 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, <a href="http://en.wikipedia.org/wiki/Grad_%28angle%29" title="Grad (angle)">grad</a>. Most computer <a href="http://en.wikipedia.org/wiki/Programming_language" title="Programming language">programming languages</a> provide function libraries that include the trigonometric functions. The <a href="http://en.wikipedia.org/wiki/Floating_point_unit" title="Floating point unit" class="mw-redirect">floating point unit</a> hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.</p> <h2><span class="editsection"></span> <span class="mw-headline" id="Applications_of_trigonometry">Applications of trigonometry</span></h2> <div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Uses_of_trigonometry" title="Uses of trigonometry">Uses of trigonometry</a></div> <p>There is an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of <a href="http://en.wikipedia.org/wiki/Triangulation" title="Triangulation">triangulation</a> is used in <a href="http://en.wikipedia.org/wiki/Astronomy" title="Astronomy">astronomy</a> to measure the distance to nearby stars, in <a href="http://en.wikipedia.org/wiki/Geography" title="Geography">geography</a> to measure distances between landmarks, and in <a href="http://en.wikipedia.org/wiki/Satellite_navigation_system" title="Satellite navigation system" class="mw-redirect">satellite navigation systems</a>. The sine and cosine functions are fundamental to the theory of <a href="http://en.wikipedia.org/wiki/Periodic_function" title="Periodic function">periodic functions</a> such as those that describe sound and <a href="http://en.wikipedia.org/wiki/Light" title="Light">light</a> waves.</p> <p>Fields which make use of trigonometry or trigonometric functions include <a href="http://en.wikipedia.org/wiki/Astronomy" title="Astronomy">astronomy</a> (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence <a href="http://en.wikipedia.org/wiki/Navigation" title="Navigation">navigation</a> (on the oceans, in aircraft, and in space), <a href="http://en.wikipedia.org/wiki/Music_theory" title="Music theory">music theory</a>, <a href="http://en.wikipedia.org/wiki/Acoustics" title="Acoustics">acoustics</a>, <a href="http://en.wikipedia.org/wiki/Optics" title="Optics">optics</a>, analysis of financial markets, <a href="http://en.wikipedia.org/wiki/Electronics" title="Electronics">electronics</a>, <a href="http://en.wikipedia.org/wiki/Probability_theory" title="Probability theory">probability theory</a>, <a href="http://en.wikipedia.org/wiki/Statistics" title="Statistics">statistics</a>, <a href="http://en.wikipedia.org/wiki/Biology" title="Biology">biology</a>, <a href="http://en.wikipedia.org/wiki/Medical_imaging" title="Medical imaging">medical imaging</a> (<a href="http://en.wikipedia.org/wiki/CAT_scan" title="CAT scan" class="mw-redirect">CAT scans</a> and <a href="http://en.wikipedia.org/wiki/Ultrasound" title="Ultrasound">ultrasound</a>), <a href="http://en.wikipedia.org/wiki/Pharmacy" title="Pharmacy">pharmacy</a>, <a href="http://en.wikipedia.org/wiki/Chemistry" title="Chemistry">chemistry</a>, <a href="http://en.wikipedia.org/wiki/Number_theory" title="Number theory">number theory</a> (and hence <a href="http://en.wikipedia.org/wiki/Cryptology" title="Cryptology" class="mw-redirect">cryptology</a>), <a href="http://en.wikipedia.org/wiki/Seismology" title="Seismology">seismology</a>, <a href="http://en.wikipedia.org/wiki/Meteorology" title="Meteorology">meteorology</a>, <a href="http://en.wikipedia.org/wiki/Oceanography" title="Oceanography">oceanography</a>, many <a href="http://en.wikipedia.org/wiki/Physical_science" title="Physical science">physical sciences</a>, land <a href="http://en.wikipedia.org/wiki/Surveying" title="Surveying">surveying</a> and <a href="http://en.wikipedia.org/wiki/Geodesy" title="Geodesy">geodesy</a>, <a href="http://en.wikipedia.org/wiki/Architecture" title="Architecture">architecture</a>, <a href="http://en.wikipedia.org/wiki/Phonetics" title="Phonetics">phonetics</a>, <a href="http://en.wikipedia.org/wiki/Economics" title="Economics">economics</a>, <a href="http://en.wikipedia.org/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>, <a href="http://en.wikipedia.org/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical engineering</a>, <a href="http://en.wikipedia.org/wiki/Civil_engineering" title="Civil engineering">civil engineering</a>, <a href="http://en.wikipedia.org/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>, <a href="http://en.wikipedia.org/wiki/Cartography" title="Cartography">cartography</a>, <a href="http://en.wikipedia.org/wiki/Crystallography" title="Crystallography">crystallography</a> and <a href="http://en.wikipedia.org/wiki/Game_development" title="Game development">game development</a>.</p> <div class="thumb tright"> <div class="thumbinner" style="width: 252px;"><a href="http://en.wikipedia.org/wiki/File:Frieberger_drum_marine_sextant.jpg" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Frieberger_drum_marine_sextant.jpg/250px-Frieberger_drum_marine_sextant.jpg" class="thumbimage" width="250" height="188" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Frieberger_drum_marine_sextant.jpg" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" alt="" width="15" height="11" /></a></div> Marine <a href="http://en.wikipedia.org/wiki/Sextant" title="Sextant">sextants</a> like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a <a href="http://en.wikipedia.org/wiki/Marine_chronometer" title="Marine chronometer">marine chronometer</a>, the position of the ship can then be determined from several such measurements.</div> </div> </div> <h2><span class="editsection"></span> <span class="mw-headline" id="Common_formulas">Common formulas</span></h2> <p>Certain equations involving trigonometric functions are true for all angles and are known as <i>trigonometric identities.</i> There are some identities which equate an expression to a different expression involving the same angles and these are listed in <a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">List of trigonometric identities</a>, and then there are the triangle identities which relate the sides and angles of a given triangle and these are listed below.</p> <div class="thumb tright"> <div class="thumbinner" style="width: 242px;"><a href="http://en.wikipedia.org/wiki/File:Triangle_ABC_with_Sides_a_b_c.png" class="image"><img alt="" src="http://upload.wikimedia.org/wikipedia/en/thumb/9/9f/Triangle_ABC_with_Sides_a_b_c.png/240px-Triangle_ABC_with_Sides_a_b_c.png" class="thumbimage" width="240" height="202" /></a> <div class="thumbcaption"> <div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Triangle_ABC_with_Sides_a_b_c.png" class="internal" title="Enlarge"><img src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" alt="" width="15" height="11" /></a></div> Triangle with sides <i>a</i>,<i>b</i>,<i>c</i> respectively opposite angles <i>A</i>,<i>B</i>,<i>C</i>, as described to the left</div> </div> </div> <p>In the following identities, <i>A</i>, <i>B</i> and <i>C</i> are the angles of a triangle and <i>a</i>, <i>b</i> and <i>c</i> are the lengths of sides of the triangle opposite the respective angles.</p> <h3><span class="editsection"></span> <span class="mw-headline" id="Law_of_sines">Law of sines</span></h3> <p>The <b><a href="http://en.wikipedia.org/wiki/Law_of_sines" title="Law of sines">law of sines</a></b> (also known as the "sine rule") for an arbitrary triangle states:</p> <dl><dd><img class="tex" alt="\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R," src="http://upload.wikimedia.org/math/6/4/3/6436a6f62b3fe9e325a1cab42d0c24fe.png" /></dd></dl> <p>where <i>R</i> is the radius of the <a href="http://en.wikipedia.org/wiki/Circumcircle" title="Circumcircle" class="mw-redirect">circumcircle</a> of the triangle:</p> <dl><dd><img class="tex" alt="R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}." src="http://upload.wikimedia.org/math/5/a/8/5a8943216b85e3eaf4c0e03a073db963.png" /></dd></dl> <p>Another law involving sines can be used to calculate the area of a triangle. If you know two sides and the angle between the sides, the area of the triangle becomes:</p> <dl><dd><img class="tex" alt="\mbox{Area} = \frac{1}{2}a b\sin C." src="http://upload.wikimedia.org/math/2/2/2/222ecf4bb160025a3cf7cebf0dbd7b64.png" /></dd></dl> <h3><span class="editsection"></span> <span class="mw-headline" id="Law_of_cosines">Law of cosines</span></h3> <p>The <b><a href="http://en.wikipedia.org/wiki/Law_of_cosines" title="Law of cosines">law of cosines</a></b> ( known as the cosine formula, or the "cos rule") is an extension of the <a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> to arbitrary triangles:</p> <dl><dd><img class="tex" alt="c^2=a^2+b^2-2ab\cos C ,\," src="http://upload.wikimedia.org/math/e/8/5/e850c08f73f03a7424cfe2ac54db3cd6.png" /></dd></dl> <p>or equivalently:</p> <dl><dd><img class="tex" alt="\cos C=\frac{a^2+b^2-c^2}{2ab}.\," src="http://upload.wikimedia.org/math/f/0/3/f0370a017db19774551a165c391ae64f.png" /></dd></dl> <h3><span class="editsection"></span> <span class="mw-headline" id="Law_of_tangents">Law of tangents</span></h3> <dl><dd><img class="tex" alt="\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}" src="http://upload.wikimedia.org/math/c/b/5/cb5c755b587d4e4e1f4367c492c4de8d.png" /></dd></dl> <h2><span class="editsection"></span></h2><h2><span class="mw-headline" id="Hubungan_fungsi_trigonometri">Hubungan fungsi trigonometri</span></h2> <dl><dd><img class="tex" alt="\sin^2 A + \cos^2 A = 1 \," src="http://upload.wikimedia.org/math/7/6/e/76ebfa9314e47d85ed1f2434ed54b471.png" /></dd></dl> <dl><dd><img class="tex" alt="1 + \tan^2 A = \frac{1}{\cos^2 A} = \sec^2 A\," src="http://upload.wikimedia.org/math/3/c/1/3c19fcedcc0320406bc77b24464aa476.png" /></dd></dl> <dl><dd><img class="tex" alt="1 + \cot^2 A = \csc^2 A \," src="http://upload.wikimedia.org/math/6/0/7/6073576f9c715f9dba3a69e72e8175dd.png" /></dd></dl> <dl><dd><img class="tex" alt="\tan A = \frac{\sin A}{\cos A}\," src="http://upload.wikimedia.org/math/e/d/9/ed97c8fb4be50976be3339e1f62b0025.png" /></dd></dl> <h2><span class="editsection"></span> <span class="mw-headline" id="Penjumlahan">Penjumlahan</span></h2> <dl><dd><img class="tex" alt="\sin (A + B) = \sin A \cos B + \cos A \sin B \," src="http://upload.wikimedia.org/math/a/2/5/a25aaac45912a2a40eacdfb969255365.png" /></dd></dl> <dl><dd><img class="tex" alt="\sin (A - B) = \sin A \cos B - \cos A \sin B \," src="http://upload.wikimedia.org/math/7/d/1/7d176b6c3427e891becb0191af26fd7c.png" /></dd></dl> <dl><dd><img class="tex" alt="\cos (A + B) = \cos A \cos B - \sin A \sin B \," src="http://upload.wikimedia.org/math/8/e/5/8e5d649a93baf1db353d78cc1bda1bd8.png" /></dd></dl> <dl><dd><img class="tex" alt="\cos (A - B) = \cos A \cos B + \sin A \sin B \," src="http://upload.wikimedia.org/math/f/3/d/f3d6075c46d48d6eb1162cb689c548c4.png" /></dd></dl> <dl><dd><img class="tex" alt="\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \," src="http://upload.wikimedia.org/math/6/3/4/63495d1b72523e5c857968de7281241f.png" /></dd></dl> <dl><dd><img class="tex" alt="\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \," src="http://upload.wikimedia.org/math/8/b/1/8b14be9a0131b9334f4ad9e1968700da.png" /></dd></dl> <h2><span class="editsection"></span> <span class="mw-headline" id="Rumus_sudut_rangkap_dua">Rumus sudut rangkap dua</span></h2> <dl><dd><img class="tex" alt="\sin 2A = 2 \sin A \cos A \," src="http://upload.wikimedia.org/math/6/b/5/6b5194dc9f8afc9cbc5a251f92cb950a.png" /></dd></dl> <dl><dd><img class="tex" alt="\cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A -1 = 1-2 \sin^2 A \," src="http://upload.wikimedia.org/math/f/1/6/f1603498de4febc0772feffc803b7ce7.png" /></dd></dl> <dl><dd><img class="tex" alt="\tan 2A = {2 \tan A \over 1 - \tan^2 A} = {2 \cot A \over \cot^2 A - 1} = {2 \over \cot A - \tan A} \," src="http://upload.wikimedia.org/math/f/4/d/f4dab83ececc78ac1c4a0ff064e2831f.png" /></dd></dl> <h2><span class="editsection"></span> <span class="mw-headline" id="Rumus_sudut_rangkap_tiga">Rumus sudut rangkap tiga</span></h2> <dl><dd><img class="tex" alt="\sin 3A = 3 \sin A - 4 \sin^3 A \," src="http://upload.wikimedia.org/math/8/a/f/8af90c1e62a9810f567bb7f654d66e9c.png" /></dd></dl> <dl><dd><img class="tex" alt="\cos 3A = 4 \cos^3 A - 3 \cos A \," src="http://upload.wikimedia.org/math/6/5/9/6599a48d2dfb0d95d02ae6980c7153a6.png" /></dd></dl> <h2><span class="editsection"></span> <span class="mw-headline" id="Rumus_setengah_sudut">Rumus setengah sudut</span></h2> <dl><dd><img class="tex" alt="\sin \frac{A}{2} = \pm \sqrt{\frac{1-\cos A}{2}} \," src="http://upload.wikimedia.org/math/a/8/2/a82000deae21e9656587d9df06dda74c.png" /></dd></dl> <dl><dd><img class="tex" alt="\cos \frac{A}{2} = \pm \sqrt{\frac{1+\cos A}{2}} \," src="http://upload.wikimedia.org/math/0/c/e/0ce4620c647b4498d835093d9cbb83eb.png" /></dd></dl> <dl><dd><img class="tex" alt="\tan \frac{A}{2} = \pm \sqrt{\frac{1-\cos A}{1+\cos A}} = \frac {\sin A}{1+\cos A} = \frac {1-\cos A}{\sin A} \," src="http://upload.wikimedia.org/math/4/4/1/44187a3edc80538d8eecd14a1bd7ecab.png" /></dd></dl><br /><p><b>Sinus</b> dalam <a href="http://id.wikipedia.org/wiki/Matematika" title="Matematika">matematika</a> adalah perbandingan sisi <a href="http://id.wikipedia.org/wiki/Segitiga" title="Segitiga">segitiga</a> yang ada di depan sudut dengan sisi miring (dengan catatan bahwa segitiga itu adalah segitiga siku-siku atau salah satu sudut segitiga itu 90<sup>o</sup>). Perhatikan segitiga di kanan; berdasarkan definisi sinus di atas maka nilai sinus adalah</p> <p><img class="tex" alt=" \sin A = {\mbox{a} \over \mbox{c}} \qquad \sin B = {\mbox{b} \over \mbox{c}}" src="http://upload.wikimedia.org/math/6/7/7/677c2eaf49d11664e0b93ca94443818c.png" /></p> <p>Nilai sinus positif di <a href="http://id.wikipedia.org/wiki/Sistem_koordinat_Kartesius" title="Sistem koordinat Kartesius">kuadran</a> I dan II dan negatif di kuadran III dan IV.</p> <h2><span class="editsection"></span><span class="mw-headline" id="Nilai_sinus_sudut_istimewa">Nilai sinus sudut istimewa</span></h2> <p><img class="tex" alt="\sin 0^o = 0\," src="http://upload.wikimedia.org/math/e/c/2/ec2d6cea315d7e55d4e9f5902755c82d.png" /></p> <p><img class="tex" alt="\sin 15^o = \frac {\sqrt{6} - \sqrt{2}}{4}\," src="http://upload.wikimedia.org/math/9/7/d/97d5b51fa76e7ecd05f2045d894cd7cc.png" /></p> <p><img class="tex" alt="\sin 30^o = \frac{1}{2}\," src="http://upload.wikimedia.org/math/7/d/a/7da718a04ba4bdd0b4888406d93a832a.png" /></p> <p><img class="tex" alt="\sin 37^o = \frac{3}{5}\," src="http://upload.wikimedia.org/math/9/5/a/95a6286091d712396666455b21ab020e.png" /></p> <p><img class="tex" alt="\sin 45^o = \frac {\sqrt{2}}{2}\," src="http://upload.wikimedia.org/math/8/2/5/825f8bbeb63946900d9a3a8932769ce2.png" /></p> <p><img class="tex" alt="\sin 53^o = \frac{4}{5}\," src="http://upload.wikimedia.org/math/0/0/2/0025b853f45af22b2fbe15c2d8443dfc.png" /></p> <p><img class="tex" alt="\sin 60^o = \frac {\sqrt{3}}{2}\," src="http://upload.wikimedia.org/math/7/1/6/716b5bc5551185f1fc9e00803f6f4ef6.png" /></p> <p><img class="tex" alt="\sin 75^o = \frac {\sqrt{6} + \sqrt{2}}{4}\," src="http://upload.wikimedia.org/math/4/2/4/424fec757261fadfb8ddaed0baf0d8d1.png" /></p> <p><img class="tex" alt="\sin 90^o = 1\," src="http://upload.wikimedia.org/math/8/d/9/8d99021f5d4752a210d3bdec010ba948.png" /></p><p><br /></p><h1 id="firstHeading" class="firstHeading">Kosinus</h1> <h3 id="siteSub">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</h3><!-- start content --> <div class="floatright"><a href="http://id.wikipedia.org/wiki/Berkas:Rtriangle.svg" class="image" title="Right triangle"><br /></a></div> <p><b>Kosinus</b> atau <i><b>cosinus</b></i> (simbol: <b>cos</b>) dalam <a href="http://id.wikipedia.org/wiki/Matematika" title="Matematika">matematika</a> adalah perbandingan sisi <a href="http://id.wikipedia.org/wiki/Segitiga" title="Segitiga">segitiga</a> yang terletak di sudut dengan sisi miring (dengan catatan bahwa segitiga itu adalah segitiga siku-siku atau salah satu sudut segitiga itu 90<sup>o</sup>). Perhatikan segitiga di kanan. Berdasarkan definisi kosinus di atas maka nilai kosinus adalah</p> <p><img class="tex" alt=" \cos A = {\mbox{b} \over \mbox{c}} \qquad \cos B = {\mbox{a} \over \mbox{c}}" src="http://upload.wikimedia.org/math/9/a/8/9a8b96952fd23b7d0c89afa09664e026.png" /></p> <p>Nilai kosinus positif di <a href="http://id.wikipedia.org/wiki/Sistem_koordinat_Kartesius" title="Sistem koordinat Kartesius">kuadran</a> I dan IV dan negatif di kuadran II dan III.</p> <h2><span class="editsection"></span> <span class="mw-headline" id="Nilai_cosinus_sudut_istimewa">Nilai cosinus sudut istimewa</span></h2> <p><img class="tex" alt="\cos 0^o = 1\," src="http://upload.wikimedia.org/math/c/4/7/c47506c6245b4b1a100a905cc538fb8e.png" /></p> <p><img class="tex" alt="\cos 15^o = \frac{\sqrt{6} + \sqrt{2}}{4}\," src="http://upload.wikimedia.org/math/a/0/6/a06ce19b9499f8ffdb0198316d80bc0d.png" /></p> <p><img class="tex" alt="\cos 30^o = \frac{\sqrt{3}}{2}\," src="http://upload.wikimedia.org/math/9/3/6/93685c7f389eb4beebb8004cce28b270.png" /></p> <p><img class="tex" alt="\cos 37^o = \frac{4}{5}\," src="http://upload.wikimedia.org/math/6/8/3/683d97b3812729afeaa83441a3b1999c.png" /></p> <p><img class="tex" alt="\cos 45^o = \frac {\sqrt{2}}{2}\," src="http://upload.wikimedia.org/math/3/4/a/34acc50a6a7561c547b30a70f50e3bcf.png" /></p> <p><img class="tex" alt="\cos 53^o = \frac{3}{5}\," src="http://upload.wikimedia.org/math/5/6/8/568900ac67465aa60e60bc02e89a5517.png" /></p> <p><img class="tex" alt="\cos 60^o = \frac {1}{2}\," src="http://upload.wikimedia.org/math/c/7/3/c73c221881defecdcf2b732306b6e5a7.png" /></p> <p><img class="tex" alt="\cos 75^o = \frac{\sqrt{6} - \sqrt{2}}{4}\," src="http://upload.wikimedia.org/math/2/d/b/2dbf14900a106a7b0598f85e1a537512.png" /></p> <p><img class="tex" alt="\cos 90^o = 0\," src="http://upload.wikimedia.org/math/7/1/2/712fe54ae1c8e356010db1c2404d00c3.png" /></p> <h2><span class="editsection"></span></h2><h1 id="firstHeading" class="firstHeading">Tangen</h1> <h3 id="siteSub">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</h3> <div id="jump-to-nav"><a href="http://id.wikipedia.org/wiki/Tangen#searchInput"><br /></a></div> <!-- start content --> <div class="floatright"><a href="http://id.wikipedia.org/wiki/Berkas:Rtriangle.svg" class="image" title="Right triangle"><br /></a></div> <p><b>Tangen</b> (<a href="http://id.wikipedia.org/wiki/Bahasa_Belanda" title="Bahasa Belanda">bahasa Belanda</a>: tangens; lambang <b>tg</b>, <b>tan</b>) dalam <a href="http://id.wikipedia.org/wiki/Matematika" title="Matematika">matematika</a> adalah perbandingan sisi <a href="http://id.wikipedia.org/wiki/Segitiga" title="Segitiga">segitiga</a> yang ada di depan sudut dengan sisi segitiga yang terletak di sudut (dengan catatan bahwa segitiga itu adalah segitiga siku-siku atau salah satu sudut segitiga itu 90<sup>o</sup>). Perhatikan segitiga di kanan; berdasarkan definisi tangen di atas maka nilai tangen adalah</p> <p><img class="tex" alt=" \tan A = {\mbox{a} \over \mbox{b}} \qquad \tan B = {\mbox{b} \over \mbox{a}}" src="http://upload.wikimedia.org/math/3/0/a/30a8ea9b4d86c5582a2cfabb5c592bb7.png" /></p> <p>Nilai tangen positif di <a href="http://id.wikipedia.org/wiki/Sistem_koordinat_Kartesius" title="Sistem koordinat Kartesius">kuadran</a> I dan III dan negatif di kuadran II dan IV.</p> <h2><span class="editsection"></span> <span class="mw-headline" id="Hubungan_Nilai_Tangen_dengan_Nilai_Sinus_dan_Cosinus">Hubungan Nilai Tangen dengan Nilai Sinus dan Cosinus</span></h2> <p><img class="tex" alt="\tan A = \frac{Sin A}{Cos A}\," src="http://upload.wikimedia.org/math/f/9/4/f9479a2492c53685173a75b7ef05575b.png" /></p> <p><br /></p> <h2><span class="editsection"></span><span class="mw-headline" id="Nilai_Tangen_Sudut_Istimewa">Nilai Tangen Sudut Istimewa</span></h2> <p><img class="tex" alt="\tan 0^o = 0\," src="http://upload.wikimedia.org/math/3/2/8/3284c4dcd4c8427f1751e42e7590c5dc.png" /></p> <p><img class="tex" alt="\tan 15^o = 2 - \sqrt {3}," src="http://upload.wikimedia.org/math/4/0/d/40d0aa264097edee90e38357eb4adec6.png" /></p> <p><img class="tex" alt="\tan 30^o = \frac{\sqrt {3}}{3}\," src="http://upload.wikimedia.org/math/9/0/6/906f2df543613d262527a1675bdfe37f.png" /></p> <p><img class="tex" alt="\tan 37^o = \frac{3}{4}\," src="http://upload.wikimedia.org/math/9/7/6/9768238cf02c318e831aa93f52c238f4.png" /></p> <p><img class="tex" alt="\tan 45^o = 1\," src="http://upload.wikimedia.org/math/a/4/6/a46337ceb904c7efa4c4e2ac10a8377c.png" /></p> <p><img class="tex" alt="\tan 53^o = \frac{4}{3}\," src="http://upload.wikimedia.org/math/c/d/4/cd4cf08c30bcb44f65b37c8c03b02412.png" /></p> <p><img class="tex" alt="\tan 60^o = \sqrt{3}\," src="http://upload.wikimedia.org/math/9/d/8/9d8ea68a7fb372ce12a4b55e2bfc9827.png" /></p> <p><img class="tex" alt="\tan 75^o = 2 + \sqrt {3}," src="http://upload.wikimedia.org/math/9/8/2/98291d958c49e6391dcf9a9f59ccfa72.png" /></p> <p><img class="tex" alt="\tan 90^o = \infty\," src="http://upload.wikimedia.org/math/0/c/2/0c2be42884c293e8c113e1a3b44c771e.png" /></p>Enyhttp://www.blogger.com/profile/14412948294694250246noreply@blogger.com0