A maths (trigo questions)

bonkysleuth SG

An isoceles triangle RST with RS = RT is inscribed in a fixed circle with centre O. The radius of the circle is 4 cm and angle SOT is 2 theta where theta is acute and measured in radians

Show that the area A cm^2 of the triangle RST is given by A = 16 sin theta + 8 sin 2 theta

Hence show that as theta varies, when the triangle RST is equilateral, triangle RST has a maximum area.

I have problems with the final part. Can someone help me out with this? thanks

Mikethm SG

A=16sin x + 8 sin 2x

dA/dx = 16cos x + 16 cos 2x

When dA/dx = 0

16cos x + 16 (2 cos^2 x -1) = 0

2cos^2 x - cos x -1 =0

(2cos x -1)( cos x + 1)=0

cos x = 1/2 or cos x = -1 (NA)

x = pi/3

theta is pi/3 and the other 2 angles same

Angles OST and OTS = pi/3

d^2A/d^2x = -16sinx - 32cos 2x

When x = pi/3

d^2A/d^2x = -ve (Max)

Therefore... blah blah blah

Done

bonkysleuth SG

H Mikethm, I tried out your method by d^2A/dx^2 is positive not negative... Thanks anyway. ^^

eagle SG

There's a careless mistake

should be

d^2A/d^2x = -16sinx - 32sin 2x

You should be able to do now :D

Mikethm SG

ya had a typo. differentiate 16 cos 2x would be -32sin 2x. Thanks.