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GOVT.SEC. SCHOOL ChhehartaGOVT.SEC. SCHOOL Chheharta

1. INTRODUCTION1. INTRODUCTION

Triangle (geometry), geometric figure consisting of three points, called vertices, connected by three sides. In Euclidean plane geometry, the sides are straight line segments (see Fig. 1). In spherical geometry, the sides are arcs of great circles (see Fig. 10). See Geometry; Trigonometry. The term triangle is sometimes used to describe a geometric figure having three vertices and sides that are arbitrary curves (see Fig. 11)

TrigonometryTrigonometry

Trigonometry, branch of mathematics that Trigonometry, branch of mathematics that deals with the relationships between the deals with the relationships between the sides and angles of triangles and with the sides and angles of triangles and with the properties and applications of the properties and applications of the trigonometric functions of angles. The two trigonometric functions of angles. The two branches of trigonometry are plane branches of trigonometry are plane trigonometry, which deals with figures lying trigonometry, which deals with figures lying wholly in a single plane, and spherical wholly in a single plane, and spherical trigonometry, which deals with triangles that trigonometry, which deals with triangles that are sections of the surface of a sphereare sections of the surface of a sphere

TYPES OF TRIANGLESTYPES OF TRIANGLES

TYPES OF TRIANGLESTYPES OF TRIANGLES

Triangles are classified in terms of their sides and angles. Scalene Triangles are classified in terms of their sides and angles. Scalene triangles have no equal sides (fig. 1), isosceles triangles have two triangles have no equal sides (fig. 1), isosceles triangles have two equal sides (fig. 4), and equilateral triangles have three equal sides equal sides (fig. 4), and equilateral triangles have three equal sides (fig. 5). In acute triangles, all the angles are less than 90° (fig. 1). In (fig. 5). In acute triangles, all the angles are less than 90° (fig. 1). In right triangles, one angle is equal to 90° (fig. 3). In obtuse triangles, right triangles, one angle is equal to 90° (fig. 3). In obtuse triangles, one angle is more than 90° (fig. 2). A line is called an altitude if it is one angle is more than 90° (fig. 2). A line is called an altitude if it is drawn from a vertex perpendicular drawn from a vertex perpendicular toto the opposite side (fig. 6). A the opposite side (fig. 6). A line is called a median if it is drawn from a vertex to the midpoint of line is called a median if it is drawn from a vertex to the midpoint of the opposite side (fig. 7). A line is called an angle bisector if it the opposite side (fig. 7). A line is called an angle bisector if it divides an angle into two equal angles (fig. 8). A line is called a divides an angle into two equal angles (fig. 8). A line is called a perpendicular bisector if it is drawn perpendicular to a side through perpendicular bisector if it is drawn perpendicular to a side through its midpoint (fig. 9). A triangle drawn on the surface of a sphere is its midpoint (fig. 9). A triangle drawn on the surface of a sphere is called a spherical triangle (fig. 10). A figure with three arbitrary called a spherical triangle (fig. 10). A figure with three arbitrary curves is sometimes called a triangle (fig. 11)curves is sometimes called a triangle (fig. 11)

II PLANE TRIANGLESII PLANE TRIANGLES

A Euclidean plane triangle has three A Euclidean plane triangle has three interior angles (the word interior is often interior angles (the word interior is often omitted), each formed by two adjacent omitted), each formed by two adjacent sides.sides.

III SPHERICAL TRIANGLESIII SPHERICAL TRIANGLES

Many properties of plane triangles have analogues in Many properties of plane triangles have analogues in spherical triangles, but the differences between the spherical triangles, but the differences between the two types are important. For example, the sum of the two types are important. For example, the sum of the angles of a spherical triangle is between 180° and angles of a spherical triangle is between 180° and 540° and varies with the size and shape of the 540° and varies with the size and shape of the triangle. A spherical triangle with one, two, or three triangle. A spherical triangle with one, two, or three right angles is called a rectangular, birectangular, or right angles is called a rectangular, birectangular, or trirectangular triangle. A spherical triangle in which trirectangular triangle. A spherical triangle in which one, two, or three sides are quadrants (quarter one, two, or three sides are quadrants (quarter circumferences) is called a quadrantal, biquadrantal, circumferences) is called a quadrantal, biquadrantal,

or triquadrantal triangleor triquadrantal triangle..

TRIGONOMETRY BUILDING TRIGONOMETRY BUILDING

Find the Height of a Building To estimate the height, H, of a building, measure the distance, D, from the point of observation to the base of the building and the angle, θ (theta), shown in the diagram. The ratio of the height H to the distance D is equal to the trigonometric function tangent θ (H/D = tan θ).

Trigonometric FunctionsTrigonometric Functions

Trigonometric functions are unitless values that Trigonometric functions are unitless values that vary with the size of an angle. An angle placed in a vary with the size of an angle. An angle placed in a rectangular coordinate plane is said to be in rectangular coordinate plane is said to be in standard position if its vertex coincides with the standard position if its vertex coincides with the origin and its initial side coincides with the positive origin and its initial side coincides with the positive x-axis.x-axis.

.. Trigonometric Identities Trigonometric Identities

The following formulas, called identities, which show the relationships between the trigonometric functions, hold for all values of the angle θ, or of two angles, θ and φ, for which the functions involved are: