Question

If x, y, z are the angles of a triangle, show that the maximum value of cos x cos y cos z is 1/8·

Answer 1

We need to find the maximum value of u = cos x cos y cos z subject to the condition x + y + z = π

Let F = cos x cos y cos z + λ (x + y + z)

We form the equations F_x = 0, F_y = 0, F_z = 0

i.e., – sin x cos y cos z + λ = 0

– cos x sin y cos z + λ = 0

– cos x cos y sin z + λ = 0

or λ = sin x cos y cos z

λ = cos x sin y cos z 

λ = cos x cos y sin z

Now, sin x cos y cos z = cos x sin y cos z = cos x cos y sin z

From the first pair, we have

sin x cos y = cos x sin y

i.e., sin x cos y – cos x sin y = 0

i.e., sin (x – y)=0 ⇒ x – y = 0

or x = y

similarly from the other pairs, we get

y = z and z = x

Combining these we have x = y = z

But x + y + z = λ

 x + x + x = π

or x = π/3

Hence x = y = z = π/3

∴ The maximum value of

cos x cos y cos z = cos3x, where x = π/3

Thus, we have cos³(π/3) = (1/2)³ = 1/8·