ظل (رياضيات)

صورة (1)

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

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

حساب الظل: في مثلث قائم الظل يساوي طول الضلع اللقاء/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 : اللقاء [أج] / المجاور [أب] 33 / 05 = 3 إذن: ظل(tan) الزاوية ب هو: 5

التاريخ

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

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**

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

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.