Solving trig function (possible error in solutions)

AI Thread Summary
The discussion revolves around a trigonometric equation where sin(2t) equals cos(2t), leading to the conclusion that tan(2t) equals 1. The participant questions the book's inclusion of both positive and negative solutions, noting that tan(-π/4) equals -1, not 1. They suggest that the problem might actually involve sin²(t) equaling cos²(t) instead. The conversation emphasizes the importance of understanding the periodic nature of trigonometric functions when determining solutions.
sandy.bridge
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Homework Statement


Alright, so I am going through the solutions to one problem that has me stumped, and initial instinct tells me it is wrong, however, I figured I would get clarification.


The Attempt at a Solution


sin(2t)=cos(2t)
tan(2t)=1
2t=\pi{/}4, 5\pi{/}4

However, the book states both positive and negative values, rather than just positive. It's either not correct, or not clicking. tan(-π/4)=-1, not 1..
 
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What is tan(-3pi/4)?
 
Since your text is suggesting that there are positive and negative solutions, you are not restricted to the "principal circle" , 0 ≤ t < 2{\pi} . So you can "go around the circle" as many times as you like:

2t = \frac{\pi}{4} + k \cdot 2{\pi} , \frac{5\pi}{4} + k \cdot 2{\pi} ,

with k being any integer.
 
sandy.bridge said:

Homework Statement


Alright, so I am going through the solutions to one problem that has me stumped, and initial instinct tells me it is wrong, however, I figured I would get clarification.


The Attempt at a Solution


sin(2t)=cos(2t)
tan(2t)=1
2t=\pi{/}4, 5\pi{/}4

However, the book states both positive and negative values, rather than just positive. It's either not correct, or not clicking. tan(-π/4)=-1, not 1..

Are you sure the problem isn't, \sin^2(t)=\cos^2(t)\,?
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

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