How to find the greatest value of this expression? "This expression" is 5

Peyton Velez

Peyton Velez

Answered question

2022-06-17

How to find the greatest value of this expression?
"This expression" is
5 sin 2 α + 4 cos 2 α 4 cos 2 β + 5 sin 2 β .
The answer is 1.25
I used simple steps to simplify this, but couldn't find the greatest value, since it has 2 kinds of angles. So how to find that?

Answer & Explanation

aletantas1x

aletantas1x

Beginner2022-06-18Added 22 answers

5 sin 2 α + 4 cos 2 α 4 cos 2 β + 5 sin 2 β = 4 + sin 2 α 4 + sin 2 β 4 + 1 4 = 5 4 .
The equality occurs for sin α = 1 and sin β = 0, which says that the answer is 5 4 ..
Done!
polivijuye

polivijuye

Beginner2022-06-19Added 16 answers

Note also that you can write your expression as
sin 2 α + 4 ( sin 2 α + cos 2 α 1 ) sin 2 β + 4 ( sin 2 β + cos 2 β 1 )
= 4 + sin 2 α 4 + sin 2 β
A ratio of positive numbers increases if the numerator increases, or if the denominator decreases. Since 0 sin 2 θ 1, this means that the largest value of the numerator is 4+1=5 and the smallest value of the denominator is 4+0=4. So the largest value of the fraction is 5 4 .

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