# Equation , range of P

#### idontknow

For which values of $P$, does the equation $$\displaystyle \sin^{10}(x)+\cos^{10}(x)=P$$ have no solutions?

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#### DarnItJimImAnEngineer

Plot P over $[-\pi,\pi]$ and see the range. My guess is the answer will be $(-\infty,0) \cup (2^{-4},\infty)$.

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#### mathman

Forum Staff
Because of tenth (even) powers, P can never be negative. The maximum is achieved when |sin(x)| or |cos(x)|=1. In that case P=1. The minimum occurs when the terms are equal or $|\sin(x)|=|\cos(x)|=\sqrt{2}/2$. In that case P$=2^{-4}$. This gives a range for P $[2^{-4},1]$.

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#### DarnItJimImAnEngineer

Well, I nearly had it, didn't I?

#### greg1313

Forum Staff
$$\displaystyle \frac{d}{dx}\left(\sin^{10}(x)+\cos^{10}(x)\right)=10\sin^9(x)\cos(x)-10\cos^9(x)\sin(x)=0\implies\sin(x)=0\,\vee\cos(x)=0\vee\sin(x)=\cos(x)$$

$$\displaystyle \Rightarrow2^{-4}\le P\le1$$

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#### skipjack

Forum Staff
The question asked for the values of $P$ for which the equation has no solutions.

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#### idontknow

The question asked for the values of $P$ for which the equation has no solutions.
Almost same question , now just avoid the interval of solutions .
$$\displaystyle P\in (-\infty , 1/16) \cup (1,\infty)$$.

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