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mathmath333
 one year ago
Trignometry question
mathmath333
 one year ago
Trignometry question

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \normalsize \text{Find the maximum value of }\hspace{.33em}\\~\\ & \sin^{4}\theta +\cos^{4}\theta \hspace{.33em}\\~\\~\\ & a.)\ 3\hspace{.33em}\\~\\ & b.)\ \dfrac13\hspace{.33em}\\~\\ & c.)\ 1 \hspace{.33em}\\~\\ & d.)\ 2 \hspace{.33em}\\~\\ \end{align}}\)

ChillOut
 one year ago
Best ResponseYou've already chosen the best response.1Rewrite either the sin or cos function. I.E \(sin^{2}(x)=1cos^{2}(x)\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \sin^{4}\theta +\cos^{4}\theta \hspace{.33em}\\~\\ & (1\cos^{2}\theta)^{2} +\cos^{4}\theta \hspace{.33em}\\~\\ \end{align}}\)

ChillOut
 one year ago
Best ResponseYou've already chosen the best response.1You now have \(cos^{2}(x)2cos^{2}(x)+cos^{4}(x)+1\), right?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \sin^{4}\theta +\cos^{4}\theta \hspace{.33em}\\~\\ &= (1\cos^{2}\theta)^{2} +\cos^{4}\theta \hspace{.33em}\\~\\ &= 12\cos^{2}\theta+2\cos^{4}\theta \hspace{.33em}\\~\\ \end{align}}\)

ChillOut
 one year ago
Best ResponseYou've already chosen the best response.1Oh, yeah, I missed a 2 there.

ChillOut
 one year ago
Best ResponseYou've already chosen the best response.1Let's call \(x=12cos^{2}(\theta)+2cos^{4}(\theta)\), so \(\frac{x}{2}=\frac{1}{2}cos^{2}(\theta)+cos^{4}(\theta)\). Well, I gotta think this for a bit (Assuming you can't use the obvious approach which is sin(x)+cos(x)=1)

ChillOut
 one year ago
Best ResponseYou've already chosen the best response.1All right... We have to rewrite that. Nevermind what I did last post. 12cos²(θ)=cos(2θ). We can work from here.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0How about using the fact that\[1^2=(\sin^2 x+\cos^2 x)^2=\sin^4 x+\cos^4 x +2 \sin^2 x \cos^2 x\]

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\sin^{4}x+\cos^{4}x=12\sin^{2}x\cos^{2}x\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and\[\text{Your expression}=1\frac{1}{2} (\sin(2x))^2\]yes the answer is 1

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4Notice \(\sin^4x\le \sin^2x\) and \(\cos^4x\le \cos^2x\). that implies \(\sin^4x+\cos^4x\le \sin^2x+\cos^2x=1\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Nice observation gane +1

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4:) i answered this problem yesterday haha!

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \text{calculate} \hspace{.33em}\\~\\ & \tan 4^{\circ} \tan 43^{\circ} \tan 47^{\circ} \tan 86^{\circ} \hspace{.33em}\\~\\ \end{align}}\) i have to solve this within a minute

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4First notice that 4+86 = 43+47 = 90

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4next recall the indentity involving tan and cot : \[\tan(x) = \cot(90x) = \dfrac{1}{\tan(90x)}\]

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2this was easy but looked hard at first sight

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4reason like this  you're supposed to solve the problem in minute, so they are "forced" to put only simple problems in the test

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4that helps in approaching the problem because when you knw upfront that they wont be asking questions that cannot be solved within a minute, you wont get distracted trying all the fancy methods..

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & \text{If}\ \sin \theta +\sin^{2} \theta =1,\ \hspace{.33em}\\~\\ & \text{and}\ \cos^{2} \theta +\cos^{4} \theta =x,\ \hspace{.33em}\\~\\ & \text{then the value of }x= \hspace{.33em}\\~\\ &a.)\ \dfrac{\cos^{2} \theta}{\sin \theta} \hspace{.33em}\\~\\ &b.)\ \text{None} \hspace{.33em}\\~\\ & c.)\ 1 \hspace{.33em}\\~\\ &d.)\ \dfrac{\sin \theta}{\cos^{2} \theta} \hspace{.33em}\\~\\ \end{align}}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4compare the first equation,\(\color{red}{\sin\theta}+\sin^2\theta=1\), with the identity : \(\color{red}{\cos^2\theta}+\sin^2\theta=1\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\sin \theta =\cos^2 \theta \)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4Yep, plug that in second equation

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & x=\sin \theta +\sin ^{2} \theta \hspace{.33em}\\~\\ \end{align}}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4\(\cos^2\theta + \cos^4\theta=x\) \(\sin\theta + \sin^2\theta=x\) \(1=x\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.2\(\large \color{black}{\begin{align} & x\cos \theta \sin \theta =1 \hspace{.33em}\\~\\ & x^{2}+(1+x^{2})\sin \theta \ \text{equals} \hspace{.33em}\\~\\ & a.)\ 0 \hspace{.33em}\\~\\ & b.)\ 2 \hspace{.33em}\\~\\ & c.)\ 1 \hspace{.33em}\\~\\ & d.)\ 1 \hspace{.33em}\\~\\ \end{align}}\)
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