Question

Let a, β, y be the three interior angles of any triangle, and try to prove that cos(a+β+y) = -1.

Answer 1

Given:

$\alpha, \beta, \gamma$ are three interior angles of any triangle.

To prove:

$\cos (\alpha+\beta+\gamma)=-1$

Solution:

According to the angle sum property of a triangle, the sum of all three interior angle of a triangle is always 180 degrees.

It is given that, $\alpha, \beta, \gamma$ are three interior angles of any triangle.

Using the angle sum property of triangles, we get

$\alpha+\beta+\gamma =180^\circ$              ...(i)

We need to prove,

$\cos (\alpha+\beta+\gamma)=-1$

Taking LHS, we get

$LHS=\cos (\alpha+\beta+\gamma)$

$LHS=\cos (180^\circ)$                [Using (i)]

$LHS=-1$

$LHS=RHS$

Hence proved.