ظل (رياضيات)

ظل الزاويه يُعرف بأنه النسبه بين الجيب وجيب التمام لنفس الزاويه.

صورة (1)

اذا نظرنا إلى صورة (1)، نرى ان المثلثات oab و OCD مماثلة، لذلك ${\frac {AB}{OA}}={\frac {DC}{OC}};{\frac {\tan x}{1}}={\frac {DC}{OC}}$ لذي ينطوي على العلاقة الاساسية بين الظل، الجيب وجيب التمام : $\tan x=\,{\frac {\sin x}{\cos x}}$

حساب الظل: في مثلث قائم الظل يساوي طول الضلع المقابل/طول الضلع المجاور

كما أن : ظل=جب/نجب مثال:

مثلا: طول الضلع [أج] =15سنتمتر طول الضلع [أب] =05سنتمتر طول الضلع [ج ب] (الوتر) =19سنتمتر لحساب ظل(tan) الزاوية ب : المقابل [أج] / المجاور [أب] 33 / 05 = 3 إذن: ظل(tan) الزاوية ب هو: 5

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 التاريخ

المقالة الرئيسية: تاريخ زاوية الظل

Function	Abbreviation	Identities (using radians)
Sine	sin	$\sin \theta \equiv \cos \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\csc \theta }}\,$
Cosine	cos	$\cos \theta \equiv \sin \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\sec \theta }}\,$
Tangent	tan (or tg)	$\tan \theta \equiv {\frac {\sin \theta }{\cos \theta }}\equiv \cot \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\cot \theta }}\,$
Cosecant	csc (or cosec)	$\csc \theta \equiv \sec \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\sin \theta }}\,$
Secant	sec	$\sec \theta \equiv \csc \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\cos \theta }}\,$
Cotangent	cot (or ctg or ctn)	$\cot \theta \equiv {\frac {\cos \theta }{\sin \theta }}\equiv \tan \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\tan \theta }}\,$

 Right triangle definitions

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 among side lengths and interior angles of a right triangle.

The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle. The number θ is the length of the curve; thus angles are being measured in radians. The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line. ("Fixed" in this context means not moving as θ changes; "moving" means depending on θ.) Thus, as θ goes from 0 up to a right angle, sin θ goes from 0 to 1, tan θ goes from 0 to ∞, and sec θ goes from 1 to ∞.

The cosine, cotangent, and cosecant functions of an angle θ constructed geometrically in terms of a unit circle. The functions whose names have the prefix co- use horizontal lines where the others use vertical lines.

The unit circle

 بعض الزوايا الشهيرة

• [إعجاز القرآن الكريم في وصف حركة الظلال http://www.nooran.org/con8/Research/8.pdf]
• ظل0=0
• ظل90=عدم تعيين
• ظل180=0
• ظل270=عدم تعيين

Trigonometric functions in the complex plane

$\sin z\,$	$\cos z\,$	$\tan z\,$	$\cot z\,$	$\sec z\,$	$\csc z\,$

 انظر أيضا

• Generating trigonometric tables
• Hyperbolic function
• Pythagorean theorem
• Unit vector (explains direction cosines)
• Table of Newtonian series
• List of trigonometric identities
• Proofs of trigonometric identities
• Euler's formula
• Polar sine — a generalization to vertex angles
• All Students Take Calculus — a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane
• Continued fraction of Gauss — a continued fraction definition for the tangent function

 وصلات خارجية

• Visionlearning Module on Wave Mathematics
• GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions
• Dave's draggable diagram. (Requires java browser plugin)

 المصادر

• Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York. (1964). ISBN 0-486-61272-4.
• Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
• Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328–339 (1996).
• Maor, Eli, Trigonometric Delights, Princeton Univ. Press. (1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.
• Needham, Tristan, "Preface"" to Visual Complex Analysis. Oxford University Press, (1999). ISBN 0-19-853446-9.
• O'Connor, J.J., and E.F. Robertson, "Trigonometric functions", MacTutor History of Mathematics Archive. (1996).
• O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2000).
• Pearce, Ian G., "Madhava of Sangamagramma", MacTutor History of Mathematics Archive. (2002).
• Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.