Sine Formula (Sine Rule)

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Theorem:

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

Or

Prove that each side of a triangle is proportional to sine of the angle between the other two i.e.,

Proof :

Let ABC be any triangle.

Draw AD ^ BC. (Produce BC if necessary in (ii))

From (1) and (2), we get

DABC is an obtuse-angled triangle.

From (4) and (5), we get

DABC is a right-angled triangle at C.

Similarly, by drawing perpendicular from B to AC, we can prove that

Hence from (6) and (9), we get

Example:

where R = Circumradius of DABC.

Suggested answer:

Fig (iii)Consider triangles (acute, obtuse and right angle) as shown in figs (i), (ii) and (iii).

Let O be the centre of the circumscribed DABC.

Let BD = 2R be the diameter of the circle. Draw CD, then we have

From the first figure, we have

From the second figure , we have

From the third figure , we have