knowledge: Applications of trigonometry

There are 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, 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.

[edit]Standard identities

Identities are those equations that hold true for any value.

$\sin^2 A + \cos^2 A = 1 \$

$\sec^2 A - \tan^2 A = 1 \$

$\csc^2 A - \cot^2 A = 1 \$

[edit]Angle transformation formulas

$\sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B$

$\cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B$

$\tan (A \pm B) = \frac{ \tan A \pm \tan B }{ 1 \mp \tan A \ \tan B}$

$\cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A }$

[edit]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.

[edit]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 and the angle between the sides, the area of the triangle is:

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

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

[edit]Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of thePythagorean theorem to arbitrary triangles:

$c^2=a^2+b^2-2ab\cos C ,\,$

or equivalently:

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

[edit]Law of tangents

The law of tangents:

$\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}$

[edit]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:

$\sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.$

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Monday, 7 January 2013

Applications of trigonometry