Home
Class 12
MATHS
Find the least value of secA+secB+secC i...

Find the least value of `secA+secB+secC` in an acute angled triangle.

Text Solution

Verified by Experts

Triangle ABC is acute angled.
`therefore 0ltA,B,Cltpi//2`
So, consider the graph of function
`y=sec x "for" x in(0,pi//2)`

The graph of function is concave upward. Now, consider three points on the graph
`P(A,secA),Q(B,secB)`
and `R(C,secC)`.
Join these points to get triangle PQR.
Centroid of the triangle PQR is
`G((A+B+C)/(3),(secA+secB+secC)/(3))`
Through point G, draw a line parallel to y-axis meeting graph to point D and x axis at point E.
The abscissa of point D is `(A+B+C)/(3)`
So, coordinates of point D are `((A+B+C)/(3),sec((A+B+C)/(3)))`
Clearly, `GEgeDE`
`therefore (secA+secB+secC)/(3)gesec((A+B+C)/(3))`
`rArr(secA+secB+secC)/(3)gesec""(pi)/(3)`
`rArr secA+secB+secCge6`
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

Find the least value of sec A+secB+secC in an acute angled triangle

The angles of a triangle are (2x)^(@), (3x + 5)^(@) and (4x − 14)^(@) . Find the value of x and the measure of each angle of the triangle

Point D,E are taken on the side BC of an acute angled triangle ABC,, such that BD = DE = EC . If angle BAD = x, angle DAE = y and angle EAC = z then the value of (sin(x+y) sin (y+z))/(sinx sin z) is ______

Let O be the circumcentre and H be the orthocentre of an acute angled triangle ABC. If A gt B gt C , then show that Ar (Delta BOH) = Ar (Delta AOH) + Ar (Delta COH)

Let A,B,C, be three angles such that A=pi/4 and tanB tanC=pdot Find all possible values of p such that A , B ,C are the angles of a triangle.

Let A,B,C, be three angles such that A=pi/4 and tanB tanC=pdot Find all possible values of p such that A , B ,C are the angles of a triangle.

If vec xa n d vec y are two non-collinear vectors and A B C isa triangle with side lengths a ,b ,a n dc satisfying (20 a-15 b) vec x+(15b-12 c) vec y+(12 c-20 a)( vec xxx vec y)=0, then triangle A B C is a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. an isosceles triangle