to find minimum value of sec &#x2061;<!-- ⁡ --> A + sec &#x2061;<!-- ⁡ --> B given

Kozankaht2

Kozankaht2

Answered question

2022-05-22

to find minimum value of sec A + sec B given A + B = π 3 , A>0 , B < π 3 .
i put B = π 3 A but how to proceed?

Answer & Explanation

Leonard Mahoney

Leonard Mahoney

Beginner2022-05-23Added 10 answers

We make use of the following identities:
( 1 ) : 2 cos A cos B = cos ( A + B ) + cos ( A B )
( 2 ) : cos A cos B + sin A sin B = cos ( A B )
Now,
sec A + sec B = 1 cos A + 1 cos B
= 1 cos A + 1 cos ( 60 A )
= cos 60 cos A + sin 60 sin A + cos A cos A cos ( 60 A )
= 3 [ 1 2 sin A + 3 2 cos A ] cos A cos ( 60 A )
= 2 3 cos ( 30 A ) 2 cos A cos ( 60 A )
(a) = 2 3 cos ( 30 A ) cos 60 + cos ( 60 2 A )
Now observe we have that if ( 30 A ) = k, then ( 60 2 A ) = 2 k. Thus, we have
( a ) = 2 3 cos k 0.5 + cos 2 k

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