Home
Class 12
MATHS
The side of a triangle inscribed in a gi...

The side of a triangle inscribed in a given circle subtends angles `alpha,betaa n dgamma` at the centre. The minimum value of the arithmetic mean of `cos(alpha+pi/2),cos(beta+pi/2)` , and `cos(gamma+pi/2)` is equal to ____

Text Solution

Verified by Experts

The correct Answer is:
`-(sqrt(3))/(2)`
Since, sides of a triangle subteands `alpha, beta, gamma` at the centra.

`:." "alpha+beta+gamma=2pi" ...(i)"`
Now, arithmetic mean
`=(cos(pi/2+alpha)+cos(pi/2+beta)+cos(pi/2+gamma))/(3)`
As we know that, `AMgeGM,` i.e.
AM is minimum, when `pi/2alpha=pi/2+beta=pi/2+gamma`
`"or "alpha=beta=gamma=120^(@)`
`:.` Minimum value of arithmetic mean
`=cos(pi/2+alpha)=cos(210^(@))=sqrt3/2`
Promotional Banner

Similar Questions

Explore conceptually related problems

If a line makes angles alpha,betaa n dgamma with threew-dimensional coordinate axes, respectively, then find the value of cos2alpha+cos2beta+cos2gammadot

alpha, beta, gamma are real number satisfying alpha+beta+gamma=pi . The minimum value of the given expression sin alpha+sin beta+sin gamma is

IF cos ( alpha + beta ) =0 then sin ( alpha + 2 beta ) is equal to

The sides of a triangle are sin alpha , cos alpha and sqrt( 1+ sin alpha cos alpha ) for some 0 lt alpha lt pi/2 then the greater angle of the triangle is

If alpha,betaa n dgamma are the roots of x^3+8=0 then find the equation whose roots are alpha^2,beta^2a n dgamma^2 .

If alpha,betaa n dgamma are roots of 2x^3+x^2-7=0 , then find the value of sum(alpha/beta+beta/alpha) .

Find the value of 2cos^3(pi/7)-cos^2(pi/7)-cos(pi/7)

Let alpha, beta, gamma be the measures of angle such that sin alpha+sin beta+sin gamma ge 2 . Then the possible value of cos alpha + cos beta+cos gamma is

The value of cos(pi/2^2).cos(pi/2^3)……….cos(pi/2^10).sin(pi/2^10) is

If the distances of the vertices of a triangle =ABC from the points of contacts of the incercle with sides are alpha,betaa n dgamma then prove that r^2=(alphabetagamma)/(alpha+beta+gamma)