Question -23
In this problem, x is allowed to be complex valued. Then find the minimum value of 2 7 cos x + 8 1 sin x .
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2 solutions
Don't we have to prove that a x that makes the equality holds is the same x that makes s i n ( x + θ ) = − 1 ? I think 2 / ( 9 ∗ 3 ) can not be achieved.
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It doesn't. I had shown this in a previous solution which was deleted by its author. You can vote up the dispute in the "View Disputes" section when you click on the "dot dot dot" at the bottom of the question, near the reshares and likes.
Why do you think that it cannot be achieved? For example, what if we set x = − 2 π − tan − 1 4 3 ?
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http://www.wolframalpha.com/input/?i=27%5E%28cos+x%29+%2B+81%5E%28sinx%29+-+%282%2F%289*root%283%29%29%29+%3D+0+real+solutions
It doesn't have real solutions. For real solutions to exist, we must have 3 c o s x = 4 s i n x and ( c o s x , s i n x ) = ( − 3 / 5 , − 4 / 5 ) , both of which can not be satisfied simultaneously.
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@Siddhartha Srivastava – Ah i see. What you meant was that "We need 3 cos x = 4 sin x and 2 7 cos x = 8 1 sin x , both of which cannot be satisfied simultaneously.
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@Calvin Lin – I messed up my initial comment, it was supposed to be ( c o s x , s i n x ) = ( − 3 / 5 , − 4 / 5 ) and not ( s i n x , c o s x ) = ( − 3 / 5 , − 4 / 5 )
The first equation I wrote is from AM-GM and the second one is to get s i n x = − 1
Here is my solution but you might need calculator for the end calculations,
To minimise the value of this expression it is obvious that the terms should be fractions, only way possible is to have negative values for both cos x and sin x which happens only in the third quadrant so later to minimise the values there we take equal angles that is 45 degrees.( Or taking 225 degrees for x at the start).
Then we find that we get: 2 7 0 . 7 0 7 1 1 + 8 1 0 . 7 0 7 1 1
Which is approximately equal to option (A).
@Sandeep Bhardwaj and @Calvin Lin comments on my solution would be helpful, just to know if the approach is correct.
Your solution is not mathematically correct. See the above solution.
- There is no reason why "minimize the values there we take equal angles".
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I came up to that conclusion after doing a lot of trial and error, I always ended up getting a higher value when I took angles other than 45. It had a huge deviation, so I thought both would be equal to 45 degrees. And I can't think of any other angle satisfying this equation, please correct me if I am wrong.
U s i n g A . M ≥ G . M . ( 2 7 c o s x + 8 1 s i n x ) / 2 ≥ 2 3 3 c o s x ∗ 3 4 s i n x = 2 3 5 ∗ ( 3 / 5 ∗ c o s x + 4 / 5 ∗ s i n x ) = 2 3 5 s i n ( x + θ ) w h e r e t a n θ = 3 / 4 S i n c e m i n ( s i n ( x + θ ) ) = − 1 S o , M i n i m u m v a l u e o f 2 7 c o s x + 8 1 s i n x = 2 / ( 9 ∗ 3 )