Relations Between Sides and Angles of a Triangles, Objectives, Sine Formula

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Objectives

After studying this lesson, you will be able to:

Sine Formula

In , the angles corresponding to the vertices and are denoted by and and the sides opposite to these vertices are denoted by and respectively. These angles and sides are called six elements of the triangle.

Prove that in any triangle, the lengths of the sides are proportional to the sines of the angles opposite to the sides,

Proof:

In in Fig. [(i), (ii), (iii)] and and is acute angle in (i), right angle in (ii) and obtuse angle in (iii).

Draw perpendicular to (or produced, if need be)

In or

In in fig (i)

In fig (ii) and

and

And fig (iii) and

Or or and

Thus, in all three figures, and

From (iii), we get

Similarly, by drawing perpendiculars from on , we can prove that

From (iv) and (v), we get

(A) is called the sine-formula

Example:

In any triangle, show that

Solution:

We have (Say)

L.H.S.

L.H.S. R.H.S.