Trigonometry

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 the trigonometric functions of those angles.]] of an angle θ can be constructed geometrically in terms of a unit circle centered at O.]] Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that deals with triangle, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

[Site] S.O.S. Mathematics: Trigonometry
Guide to trigonometric functions, equations, and identities.
www.sosmath.com/trig/trig.html

Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called “trig.”

[News] Student spotlight: Kyle Shanda
• Age: 17

A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.

[Image]
Straight Forward Math Series: Pre-Calculus $9.95 Straight Forward Math Series: Trigonometry $9.95

History

]]Trigonometry was probably developed for use in sailing as a navigation method used with astronomy. The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly computes the sine of $\frac{\pi}{4}$ (=45°) as $\frac{1}{\sqrt{2}}$ in a procedure for "circling the square" (i.e., constructing the inscribed circle).

[Video] Physics: Vector components (26)

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus[1] circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD.

[Auction] Sullivan Algebra & Trigonometry Student Solutions Manua
Only $10.0

The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC. LankaLibrary Forum :: View topic - Oldest documented Hospital system (4 B.C) in Sri Lanka

[Post] Trigonometry Workbook For Dummies
No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen ...

The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.

[Book] Trigonometry For Dummies (For Dummies (Math & Science)) For Dummies

In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical geometry:

[Site] Trigonometry -- from Wolfram MathWorld
Trigonometry ... Weisstein, E. W. "Books about Trigonometry." http://www.ericweisstein.com/encyclopedias/books/Trigonometry.html. ...
mathworld.wolfram.com/Trigonometry.html

\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.

[News] Prep Athlete of the Week: Diaz contributes with versatility
HAMILTON CITY — Omar Diaz engineered himself quite the stat line Friday. Maybe in the future he'll be engineering bigger things.

Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula

[Image]
Super Tutor Trigonometry Click for the Back of the Box

\cos a \cos b = \frac{\cos(a+b) + \cos(a-b)}{2}.

[Video] Physics: Vector components (25)

Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry. Lagadha (circa 1350-1200 BC) is the first person thought to have used geometry and trigonometry for astronomy, in his Vedanga Jyotisha.

[Auction] TRIGONOMETRY AND PRE-CALCULUS TUTOR: - DVD BOXSET *NEW
Only $29.5

Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation $x^3 + 200 x = 20 x^2 + 2000$ and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables.

[Post] Trigonometry Workbook For Dummies
Trigonometry Workbook For DummiesGot a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where...

Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry.

[Book] Algebra and Trigonometry: Structure and Method Book 2 Houghton Mifflin Company

The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.

[Site] Dave's short course in trigonometry
Covers basic definitions and concepts in beginning trigonometry. ... Trigonometry began with chords. Sines. The relation between sines and chords. The word "sine" ...
www.clarku.edu/~djoyce/trig

In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy.

[News] CENTRAL NEW YORK NEWS
Tag your photos 'news' to appear on this page.

The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry".

[Image]
Trigonometric Table - Jpeg Image (STUDY THIS CAREFULLY!!)

Overview

If one angle of a right 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 right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. 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:

[Video] Trigonometry

The 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\,}\,.

The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.

The 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}\,.

[Auction] NEW Algebra and Trigonometry Enhanced Graphing Utili...
Only $149.33

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).

[Post] Afterimages of TANATOS : "trigonometry"
TommyOshima posted a photo:. Afterimages of TANATOS : "trigonometry". With Zeissism, Kinako Kocteau, Velco in the streets of OSAKA.

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

[Book] Trigonometry Workbook For Dummies (For Dummies (Lifestyles Paperback)) For Dummies

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 an 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.

[Site] trigonometry: Definition from Answers.com
trigonometry n. The branch of mathematics that deals with the relationships between the sides and the angles of triangles and the calculations based ... Trigonometry ...
www.answers.com/topic/trigonometry

Extending the definitions

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.

[News] Charles W. Fredlund
After moving to Baltimore to build warplanes, he took a job as a machinist while building his house in Carney by hand. C harles William "Smiling Charlie" Fredlund, a retired machinist, died Tuesday of cancer in the Carney home he designed and built with his own hands more than 50 years ago. He was 84.

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 function cis is particularly useful

[Image]
To better understand certain problems involving aircraft and propulsion it is necessary to use some mathematical ideas from trigonometry , the study of triangles. Most people are

\operatorname{cis}\,x = \cos x + i\sin x \! = e^{ix}.

[Video] Section 4.4 Problem 20 Example

See Euler's and De Moivre's formulas.

[Auction] ALGEBRA TRIGONOMETRY GRAPHS 3RD 2006 HB 0321279115 NM
Only $48.0

Mnemonics

[Book] Algebra and Trigonometry, 6th Edition Brooks Cole

Students often use mnemonics 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 as strings of letters, as in SOH-CAH-TOA.

[Site] Algebra II: Trigonometry - Math for Morons Like Us
Math for Morons Like Us - Algebra II: Trigonometry ... Trigonometry. Trig. Identities. Equations & Tri. ... Take the Quiz on trigonometry. ...
library.thinkquest.org/20991/alg2/trig.html

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

[News] Crises help create a teachable moment
U nemployment is rising. The stock market is setting all sorts of records - mostly bad. And the economy might be headed into a prolonged recession.

Alternatively, one can devise sentences which consist of words beginning with the letters to be remembered. For example, to recall that Tan = Opposite/Adjacent, the letters T-O-A must be remembered. Any memorable phrase constructed of words beginning with the letters T-O-A will serve.

[Image]
. The new edition of Trigonometry has been published. For those of you using the book, you will notice that I have a coauthor, Mark Turner, for the first time. Mark has been a colleague of mine at Cuesta College for a number of years, and I am very pleased to have the opportunity to work with him on this edition. He is an excellent teacher

Another type of mnemonic describes facts in a simple, memorable way, such as "Plus to the right, minus to the left; positive height, negative depth," which refers to trigonometric functions generated by a revolving line.

[Video] Section 4.4 Problem 15 Example

Calculating trigonometric functions

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.

[Auction] ALGEBRA & TRIGONOMETRY 7TH 2005 HB 1033PGS 0131430734 G
Only $38.0

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 have built in instructions for calculating trigonometric functions.

[Post] Simple Trigonometry in XNA
Johny Gizmo has a new XNA tutorial up focusing on basic trigonometry. This tutorial covers angles and how to work with the sine, cosine and tangent angles. Read More Johnny Gizmo ...

Applications of trigonometry

There are an enormous number of applications 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.

[Book] Trigonometry (Lial/Hornsby/Schneider Series) Addison Wesley

Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, 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, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

[Site] Trigonometry
A trigonometry tutorial which aims to communicate the point of trigonometry, as well as the basics of the subject - with links to definitions and trig-based animations.
oolong.co.uk/trig.htm

like this 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 then be determined from several such measurements.]]

[News] Flush from a loo dream
I have a dream. A dream that one day, every bathroom in our land will be clean. Where every citizen, regardless of race, creed, caste or religion, will know how to use and flush a toilet.

Common formulae

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships. For example, the Pythagorean identities are an expression of the Pythagorean Theorem. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see trigonometric identity.

[Image]
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships between side lengths and interior

Trigonometric identities

Pythagorean identities

\begin{align}

\sin^2 \alpha + \cos^2 \alpha = 1 \\\tan^2 \alpha + 1 = \sec^2 \alpha \\1+\cot^2 \alpha = \csc^2 \alpha \end{align}

[Video] Section 4.4 Problem 4 Example

Sum and product identities

Sum to product:

\begin{align}

\sin \alpha \pm \sin \beta &= 2\sin \left( \frac{\alpha \pm \beta}{2}\right)\cos \left(\frac{\alpha \mp \beta}{2} \right)\\\cos \alpha + \cos \beta &= 2\cos \left(\frac{\alpha + \beta}{2} \right)\cos \left(\frac{\alpha - \beta}{2}\right)\\\cos \alpha - \cos \beta &= -2\sin \left(\frac{\alpha + \beta}{2} \right) \sin \left(\frac{\alpha - \beta}{2}\right)\end{align}

[Auction] TRIGONOMETRY/TRIG MATH DVD OR INSTANT ACCESS VIDEO
Only $49.99

Product to sum:

\begin{align}

\cos \alpha \,\cos \beta &= \frac{1}{2} - \beta) + \cos (\alpha + \beta)\\\sin \alpha \,\sin \beta &= \frac{1}{2} - \beta) - \cos (\alpha + \beta)\\\cos \alpha \,\sin \beta &= \frac{1}{2} + \beta) - \sin (\alpha - \beta)\\\sin \alpha \,\cos \beta &= \frac{1}{2} + \beta) + \sin (\alpha - \beta)\end{align}

[Post] UNsolved problem 164121 added to Trigonometry-basics
Click on the link to solve this problem: 3+2i/8-2i.

Sine, cosine, and tangent of a sum

Detailed, diagramed proofs of the first two of these formulasare available
for download as a four-page PDF documentat .

[Book] Functions, Statistics, and Trigonometry (The University of Chicago School Mathematics Project) Scott Foresman & Co

\begin{align}

\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\\cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\\tan(\alpha \pm \beta) &= \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}\end{align}

[Site] Trigonometry - Wikiversity
Wikibooks has a book on the topic of Trigonometry. ... The insights of trigonometry permeate other branches of geometry, such as the ...
en.wikiversity.org/wiki/Trigonometry

Half-angle identities

Note that

\pm

is correct, it means it may be either one, depending on the value of A/2.

\begin{align}

\sin \frac{A}{2} &= \pm \sqrt{\frac{1-\cos A}{2}} \\\cos \frac{A}{2} &= \pm \sqrt{\frac{1+\cos A}{2}} \\\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}\end{align}

[Image]
toilet_flushing.mp3 21-Aug-2004 13:04 284K tomorrow.jpg 07-Feb-2005 14:52 12K trigonometry.jpg 07-Apr-2005 07:05 11K turducken.jpg 02-Dec-1999 20:45 13K

Stereographic ( or parametric ) identities

\begin{align}

\sin \alpha &= \frac{2T}{1+T^2} \\\cos \alpha &= \frac{1-T^2}{1+T^2}\\\end{align}

[Video] Section 4.4 Problem 3 Example

where $T=\tan \frac{\alpha}{2}$ .

[Auction] Vintage Trigonometry Tables by Illinois Tool Works
Only $0.99

Triangle identities

[[Image:Triangle ABC with Sides a b c.png|thumb|240px|Laws of Sines and Cosines

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

[Post] UNsolved problem 164129 added to Trigonometry-basics
Click on the link to solve this problem: Which of the following is equivalent to 1 - cos2 - 0 ------------ ??? AND WHY? cos2 - 0 A = TAN2 0 B = SIN2 0 C - 1 ------ SIN2 0 Thanks.

$\cos C=\frac{a^2+b^2-c^2}{2ab}$ ]]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 know as the "sine rule") for an arbitrary triangle states:

[Book] College Algebra and Trigonometry (4th Edition) (Mathxl Tutorials on CD) Addison Wesley

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,

where R is the radius of the circumcircle of the triangle.

[Site] Trigonometry - Wikibooks, collection of open-content textbooks
Core concepts of trigonometry. Prerequisites and Basics. In simple terms ... Right angle trigonometry. Trigonometric identities. Graphs of Sine and Cosine Functions ...
en.wikibooks.org/wiki/Trigonometry

Law of cosines

The law of cosines (also 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 ,\,

[Image]
doing it on the computer wouldn't be as effective. I have done pages on polynomials, on conic sections, on formulas for describing acceleration, and on trigonometry. Here is an image of my latest page showing the formula for the trajectory of a projectile fired at an angle. The arrangement is mine, but the sequence of information comes from one of my trigonometry books. I keep

or equivalently:

[Video] Section 4.4 Problem 1 Example

\cos C=\frac{a^2+b^2-c^2}{2ab}.\,

[Auction] TRIGONOMETRY UNIT CIRCLE APPROACH 7TH 2005 0131431110 G
Only $33.0

Law of tangents

The law of tangents:

[Post] Trigonometry
why is ? i'm really stumped on this one. Thanks EDIT: fixed the latex.

\frac{a+b}{a-b}=\frac{\tan\left [2]}{\tan\left [3]}

[Book] Algebra and Trigonometry: An Early Functions Approach (2nd Edition) Prentice Hall

[Image]

[Video] Physics: Vector components (24)

[Auction] Analytic Trigonometry with Applications by Karl E. B...
Only $15.0

[Post] Limits, L'Hospital's Rule & Trigonometry
1. The problem statement, all variables and given/known data The question asks: Find the limit. Use L'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If L'Hospital's Rule doesn't apply, ...

[Book] Algebra and Trigonometry (4th Edition) Prentice Hall

[Image]
attime.jpg 13-Nov-2003 06:52 35K iNotePad.jpg 25-May-2002 00:56 32K trigonometry.jpg 16-Jan-2005 07:52 27K xfi_large.jpg 25-May-2002 00:56 23K

[Video] Physics: Vector components (23)

[Auction] GEOMETRY AND TRIGONOMETRY 1872 CHARLES DAVIES FR SHIPIN
Only $9.0

[Post] Trigonometry
Heres a complete, easy-to-grasp course in trigonometry that takes the form of a fantasy novel. The King of Carmorra and his subjects have many practical problems to solve, and their answers can be found by applying principles of ...