Jumat, 15 Januari 2010

rigonometry (from Greek trigōnon "triangle" + metron "measure")[1] is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships, as well as describing angles in general and the motion of waves such as sound and light waves.

Trigonometry is usually taught in secondary schools either as a separate course or as part of a precalculus course. It has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.

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 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:

  • 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}\,.

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. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

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.

Extending the definitions

Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians.

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 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

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

See Euler's and De Moivre's formulas.


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 as strings of letters, as in SOH-CAH-TOA.

Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent

The memorization of this mnemonic can be aided by expanding it into a phrase, such as "Some Officers Have Curly Auburn Hair Till Old Age".[4] Any memorable phrase constructed of words beginning with the letters S-O-H-C-A-H-T-O-A will serve.

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.

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.

Applications 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 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.

Marine sextants 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.

Common formulas

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. There are some identities which equate an expression to a different expression involving the same angles and these are listed in List of trigonometric identities, and then there are the triangle identities which relate the sides and angles of a given triangle and these are listed below.

Triangle with sides a,b,c respectively opposite angles A,B,C, as described to the left

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 circumcircle 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. If you know two sides and the angle between the sides, the area of the triangle becomes:

\mbox{Area} = \frac{1}{2}a b\sin C.

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}.\,

Law of tangents


Hubungan fungsi trigonometri

\sin^2 A + \cos^2 A = 1 \,
1 + \tan^2 A = \frac{1}{\cos^2 A} = \sec^2 A\,
1 + \cot^2 A = \csc^2 A \,
\tan A = \frac{\sin A}{\cos A}\,


\sin (A + B) = \sin A \cos B + \cos A \sin B \,
\sin (A - B) = \sin A \cos B - \cos A \sin B \,
\cos (A + B) = \cos A \cos B - \sin A \sin B \,
\cos (A - B) = \cos A \cos B + \sin A \sin B \,
\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \,
\tan (A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \,

Rumus sudut rangkap dua

\sin 2A = 2 \sin A \cos A \,
\cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A -1 = 1-2 \sin^2 A \,
\tan 2A = {2 \tan A \over 1 - \tan^2 A} = {2 \cot A \over \cot^2 A - 1} = {2 \over \cot A - \tan A} \,

Rumus sudut rangkap tiga

\sin 3A = 3 \sin A - 4 \sin^3 A \,
\cos 3A = 4 \cos^3 A - 3 \cos A \,

Rumus setengah sudut

\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} \,

Sinus dalam matematika adalah perbandingan sisi segitiga yang ada di depan sudut dengan sisi miring (dengan catatan bahwa segitiga itu adalah segitiga siku-siku atau salah satu sudut segitiga itu 90o). Perhatikan segitiga di kanan; berdasarkan definisi sinus di atas maka nilai sinus adalah

 \sin A = {\mbox{a} \over \mbox{c}} \qquad \sin B = {\mbox{b} \over \mbox{c}}

Nilai sinus positif di kuadran I dan II dan negatif di kuadran III dan IV.

Nilai sinus sudut istimewa

\sin 0^o = 0\,

\sin 15^o = \frac {\sqrt{6} - \sqrt{2}}{4}\,

\sin 30^o = \frac{1}{2}\,

\sin 37^o = \frac{3}{5}\,

\sin 45^o = \frac {\sqrt{2}}{2}\,

\sin 53^o = \frac{4}{5}\,

\sin 60^o = \frac {\sqrt{3}}{2}\,

\sin 75^o = \frac {\sqrt{6} + \sqrt{2}}{4}\,

\sin 90^o = 1\,


Dari Wikipedia bahasa Indonesia, ensiklopedia bebas

Kosinus atau cosinus (simbol: cos) dalam matematika adalah perbandingan sisi segitiga yang terletak di sudut dengan sisi miring (dengan catatan bahwa segitiga itu adalah segitiga siku-siku atau salah satu sudut segitiga itu 90o). Perhatikan segitiga di kanan. Berdasarkan definisi kosinus di atas maka nilai kosinus adalah

 \cos A = {\mbox{b} \over \mbox{c}} \qquad \cos B = {\mbox{a} \over \mbox{c}}

Nilai kosinus positif di kuadran I dan IV dan negatif di kuadran II dan III.

Nilai cosinus sudut istimewa

\cos 0^o = 1\,

\cos 15^o = \frac{\sqrt{6} + \sqrt{2}}{4}\,

\cos 30^o = \frac{\sqrt{3}}{2}\,

\cos 37^o = \frac{4}{5}\,

\cos 45^o = \frac {\sqrt{2}}{2}\,

\cos 53^o = \frac{3}{5}\,

\cos 60^o = \frac {1}{2}\,

\cos 75^o = \frac{\sqrt{6} - \sqrt{2}}{4}\,

\cos 90^o = 0\,


Dari Wikipedia bahasa Indonesia, ensiklopedia bebas

Tangen (bahasa Belanda: tangens; lambang tg, tan) dalam matematika adalah perbandingan sisi segitiga 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 90o). Perhatikan segitiga di kanan; berdasarkan definisi tangen di atas maka nilai tangen adalah

 \tan A = {\mbox{a} \over \mbox{b}} \qquad \tan B = {\mbox{b} \over \mbox{a}}

Nilai tangen positif di kuadran I dan III dan negatif di kuadran II dan IV.

Hubungan Nilai Tangen dengan Nilai Sinus dan Cosinus

\tan A = \frac{Sin A}{Cos A}\,

Nilai Tangen Sudut Istimewa

\tan 0^o = 0\,

\tan 15^o = 2 - \sqrt {3},

\tan 30^o = \frac{\sqrt {3}}{3}\,

\tan 37^o = \frac{3}{4}\,

\tan 45^o = 1\,

\tan 53^o = \frac{4}{3}\,

\tan 60^o = \sqrt{3}\,

\tan 75^o = 2 + \sqrt {3},

\tan 90^o = \infty\,