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# DEFINITION OF THE SIX TRIGONOMETRIC 11

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       13/07/2018 14:15:10
The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.

To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.
In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° (? radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (
?
/
2
radians), so each of these angles must be in the range of (0,
?
/
2
) as expressed in interval notation. The following definitions apply to angles in this (0,
?
/
2
) range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin(?) for angles ?, ? ? ?, ? + ?, and 2? ? ? depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of ? is extended to additional rotations, this behavior repeats periodically with a period 2?.

The trigonometric functions are summarized in the following table and described in more detail below. The angle ? is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram.

المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .