Trigonometry #6: Find the Error
by djmathman, Nov 7, 2011, 2:08 AM
I have a proof for this problem, but I've been trying to find a second proof. It seems correct (and the answer is indeed correct), but I noticed an error somewhere in this proof. Can you find the error and help me correct it? Thanks. (I know where the error is; I just don't know how to fix it)
We look to the unit circle for inspiration. We see that the maximum will occur when
and 
are both positive, so we focus on the first quadrant. For any right triangle in the unit circle, with leg lengths 
and 
, we are given that 
since the radius of the circle is equal to 
. Using 
on 
and 
we find that 
, or 
. Since we know that , we can substitute to get that 
.
Now we use one last time, this time on 
and 
. Doing so, we get 
. Substituting our range for 
, we get 
, or 
. Finally, multiplying both sides of the inequality by 
gives us the desired 
.
Problem wrote:
Find the maximum value of 
for 
.
for 
.We look to the unit circle for inspiration. We see that the maximum will occur when
and 
are both positive, so we focus on the first quadrant. For any right triangle in the unit circle, with leg lengths 
and 
, we are given that 
since the radius of the circle is equal to 
. Using 
on 
and 
we find that 
, or 
. Since we know that , we can substitute to get that 
.Now we use
one last time, this time on 
and 
. Doing so, we get 
. Substituting our range for 
, we get 
, or 
. Finally, multiplying both sides of the inequality by 
gives us the desired 
.
.
.
,
.
,
April 2025
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