Home
Class 12
MATHS
Let alpha,betaa n dgamma satisfy 0 < alp...

Let `alpha,betaa n dgamma` satisfy `0 < alpha < beta < gamma < 2pidot` It `cos(x+alpha)+cos(x+beta)+cos(x+gamma)=0` for all`x in R ,` then find the possible values of `(gamma-alpha)`

Text Solution

Verified by Experts

Given, `cos(x+alpha)+cos(x+beta)+cos(x+gamma)=0AAx in R`
`therefore cos x(cos alpha+cosbeta+cos gamma)-sin x(sin alpha+sin beta+sin gamma)`
`=0 AA x in R`
`therefore cos alpha+cos beta+cos gamma=0=sin alpha+sin beta+sin gamma`
`therefore cos alpha+cos gamma=-cos beta`
and `sin alpha+sin gamma=-sin beta`
Squaring Eqs. 1 and 2 and then adding then, we get
`2+2cos(alpha-gamma)=1`
`rArr cos (gamma-alpha)=(-1)/(2)`
`rArr gamma-alpha=(2pi)/(3)` or `(4pi)/(3)`
Promotional Banner

Topper's Solved these Questions

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.1|11 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.2|7 Videos

  • TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise SINGLE CORRECT ANSWER TYPE|38 Videos

  • TRIGONOMETRIC RATIOS FOR COMPOUND, MULTIPLE, SUB-MULTIPLE ANGLES, AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Multiple Correct Answers Type|6 Videos

Similar Questions

Explore conceptually related problems

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

Let alpha,betaa n dgamma be some angles in the first quadrant satisfying tan(alpha+beta)=(15)/8a n dcos e cgamma=(17)/8, then which of the following hold(s) good? alpha+beta+gamma=pi cotalpha+tanbeta+tangamma=tanalphatanbetatangamma tanalpha+tanbeta+tangamma=tanalphatanbetatangamma tanalphatanbeta+tanbetatangamma+tangammatanalpha=1

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 ____

A tower P Q stands at a point P within the triangular park A B C such that the sides a , ba n dc of the triangle subtend equal angles at P , the foot of the tower. if the tower subtends angles alpha,betaa n dgamma,a tA , Ba n dC respectively, then prove that a^2(cotbeta-cotgamma)+b^2(cotgamma-cotalpha)+a^2(cotalpha-cotbeta)=0

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 .

A line makes angles alpha,betaa n dgamma with the coordinate axes. If alpha+beta=90^0, then find gammadot

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)

Let x-ysinalpha-zs inbeta=0,xs inalpha=zs ingamma-y=0a n dxsinbeta+ysingamma-z=0 be the equations of the planes such that alpha+beta+gamma=pi//2(w h e r ealpha,betaa n dgamma!=0)dot Then show that there is a common line of intersection of the three given planes.

A right angle is divided into three positive parts alpha,betaa n dgammadot Prove that for all possible divisions t a nalpha+tanbeta+tangamma>1+tanalphatanbetatangammadot

If alpha,beta,a n dgamma are the an gles which a directed line makes with the positive directions of the co-ordinates axes, then find the value of sin^2alpha+sin^2beta+sin^2gammadot