Relations Between Sides and Angles of a Triangles, Cosine Formula

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Cosine Formula

In any triangle, prove that

(i)

(ii)

(iii)

Proof:

Cosine Formula

Cosine Formula

Cosine Formula

Three cases arise:

(i) When is acute

(ii) When is a right acute

(iii) When is obtuse

Let us consider these one by one:

Case: (i)

When is acute,

Also

From fig (i)

Case (ii)

When

Case (iii)

When is obtuse

Also,

In all the three cases,

Similarly, it can be proved that and

Example 1: In any triangle, show that

Solution:

We know that

L.H.S.

R.H.S.

Example 2: If the sides of a triangle are and find the greatest angle of the triangle.

Solution:

Here

We know that in a triangle, the angle opposite to the largest side is greatest

is the greatest angle

The greatest angle of the triangle is or

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