Find the minimum value of the trigonometric equation If x+y=2c, find minimum value of secx+secy if x,y in (0,pi/2), in terms of c. I was able to solve by differentiating the equation and got the answer as 2secc. But i would like to know solution with trigonometry as base or without differentiating the equation.

reevelingw97

reevelingw97

Answered question

2022-11-06

Find the minimum value of the trigonometric equation
If x + y = 2 c, find minimum value of sec x + sec y if x , y ( 0 , π / 2 ), in terms of c
I was able to solve by differentiating the equation and got the answer as 2secc. But i would like to know solution with trigonometry as base or without differentiating the equation.

Answer & Explanation

Eva Cochran

Eva Cochran

Beginner2022-11-07Added 14 answers

Here is a method without differentiation.
Since x , y ( 0 , π / 2 )
sec x , sec y > 0
By using A . M G . M, we have
sec x + sec y 2 sec x sec y
sec 2 x + sec 2 y + 2 sec x sec y 4 sec x sec y
sec 2 x + sec 2 y 2 sec x sec y 0
( sec x sec y ) 2 0
sec x sec y = 0
x = y
Since, x + y = 2 c
x + x = 2 c
x = c
sec x + sec y = 2 sec x  which is the minimum
Barrett Osborn

Barrett Osborn

Beginner2022-11-08Added 3 answers

d d x ( sec x + sec y ) = sin x cos 2 x + sin y cos 2 y d y d x = 0 , d y d x = 1 , sin x cos 2 x = sin y cos 2 y x = y = c .
Minimum value: 2 sec c .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?