1. ## Trig Optimization Problem

I am having a bit of trouble getting started with this problem:
The hypotenuse of a right-angled triangle is 12 cm in length. Find the measures of the angles in the triangle that maximize its perimeter.
Any help would be appreciated.

2. ## calculus

It's a right triangle. So x^2 + y^2 = 12^2. You want to maximize the perimenter = x+y+12. After you have x and y, you can use trig to find the angles

3. ## Re: Trig Optimization Problem

Right Angle Triangle

Perimeter = Side1 + S2 + S3

Side 1 = 12cm (hypotenuse)

S2 = 12cosX (if you draw a right angled triangle you can figure this out through basic trig)
S3 = 12sinX

Therefore;

P(x) = Side1 + S2 + S3
P(x) = 12 + 12CosX + 12SinX

To find max; use calculus!

P'(x) = 12(-sinX) + 12(cosX)
P'(x) = 12(cosX - sinX)

Max when this = 0

P'(x) = 0
12(cosX - sinX) = 0
therefore; cosX - sinX = 0
therefore; cosX = sinX

Only place these are = to eachother is at 45 degrees, or Pi/4 radians.

To check this is a max. Upon graphing 0 < x < 90 (angle cannot be less than or zero, or cannot be more than or equal to 90 degrees)

Graph

0_______ (positive)_________45 (0)_______________(negative)_____________90

transitions from + to - therefore Perimeter is maximised at 45 degrees.

Hope this helped