Home
Class 12
MATHS
In A B C Prove that cos^2A/2+cos^2B/2+...

In ` A B C` Prove that `cos^2A/2+cos^2B/2+cos^2C/2lt=9/4dot` In `cos^2A/2+cos^2B/2+cos^2C/2=y(x^2+1/(x^2))` then find the maximum value of `ydot`

Text Solution

Verified by Experts

The correct Answer is:
`9//8`
(a) In `triangle ABC`, we know that
`cosA+cosB+cos Cle(3)/(2)`
`cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)`
`=(1+cosA)/(2)+(1+cosB)/(2)+(1+cosC)/(2)`
`=(3)/(2)+(cos A+cosB+cosC)/(2)le(3)/(2)+(3)/(4)` (using eq. i)
`therefore cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)le(9)/(4)`.
(b) `cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)=y(x^(2)+(1)/(x^(2)))`
`therefore y(x^(2)+(1)/(x^(2)))le(9)/(4)`.
`therefore yle(9)/(4(x^(2)+(1)/(x^(2)))`
Now `x^(2)+(1)/(x^(2))ge2`
`therefore yle(9)/(8)`.
thus, maximum value of `y` is `(9)/(8)`.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise (Single)|100 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise (Multiple)|22 Videos

  • TRIGONOMETRIC RATIOS AND TRANSFORMATION FORMULAS

    CENGAGE|Exercise Exercise 3.9|5 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 A+B+C=180^0 , prove that : cos^2, A/2 + cos^2, B/2 - cos^2, C/2 = 2cos, A/2 cos, B/2 sin, C/2

In A B C , prove that (a-b^2cos^2C/2+(a+b)^2sin^2C/2=c^2dot

Knowledge Check

  • IF A + B + C = pi then cos ^2 A + cos^2 B + cos^2 C=

    A
    ` 1- cos A cos B cos C`
    B
    ` 1- 2 cos A cos B cos C `
    C
    `2 cos A cos B cos C `
    D
    `1+ cos A cos B cos C `

    • Similar Questions

    Explore conceptually related problems

    Prove that sin^2 A cos^2 B+cos^2 A sin^2 B+cos^2 A cos^2 B+sin^2 A sin^2 B=1

    If A+B+C = pi show that cos^2 (A/2)-cos^2 (B/2)-cos^2 (C/2)=-2sin(A/2)cos(B/2)cos(C/2)

    In A B C , prove that cos e c A/2+cos e c B/2+cos e c C/2geq6.

    cos^2 A + cos^2 B +cos^2 C=1+2cosA cosB cosC .

    If A+B+C=pi prove that cos^(2) A+cos^(2) B+cos^(2) C=1 - 2cos A cos B cos C .

    If A+B+C=pi prove that cos^(2)A+cos^(2)B+cos^(2)C=1-2cosAcosBcosC .

    In triangleABC , prove that cos^(2)A+cos^(2)B+cos^(2)Cge(3)/(4) .

    Home
    Profile