Number of Solutions of $\sin^4 x+\cos^7 x=1$

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  • Thread starter juantheron
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In summary, the equation $\sin^4 x+\cos^7 x=1$ has at least two solutions, which can be found by solving for $\cos x$ and setting it equal to $\pm \frac{\pi}{2}$. By substitution and simplification, the equation can be reduced to $t^5+t^2-2=0$, where $t=\cos x$. This equation has a maximum of two solutions, which can be found by setting $t^5=t^2=1$.
  • #1
juantheron
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Total number of solution of $\sin^4 x+\cos^7 x= 1\;,$ Where $x\in \left[-\pi,\pi\right]$
 
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  • #2
Can you write a lower bound on the number of solutions?
 
  • #3
Yes Evgeny Makarov,

$\sin^4x+\cos^7x = 1\Rightarrow \cos^7 x = 1-\sin^4 x= \cos^2 x\cdot (1+\sin^2 x)$

So $\cos^2 x \cdot \left[\cos^5 x-1-\sin ^2 x\right] = 0$

So either $\cos^2 x=0$ or $\cos^5 x = 1+\sin^2 x$

So we get $\displaystyle x=\pm \frac{\pi}{2}$ Now how can i solve after that, Help me

Thanks
 
  • #4
jacks said:
So either $\cos^2 x=0$ or $\cos^5 x = 1+\sin^2 x$

So we get $\displaystyle x=\pm \frac{\pi}{2}$ Now how can i solve after that
Denote $t=\cos x$; then $\sin^2x=1-t^2$ and the equation becomes $t^5+t^2-2=0$. Note that $t\le 1$, so $t^2\le1$ and $t^5\le1$. Therefore, $t^5+t^2-2\le0$ and $t^5+t^2-2=0$ iff $t^5=t^2=1$. Can you finish?
 

FAQ: Number of Solutions of $\sin^4 x+\cos^7 x=1$

How many solutions does the equation sin4x + cos7x = 1 have?

The equation has an infinite number of solutions.

Is there a general formula for finding the solutions of this equation?

No, there is no general formula for finding the solutions of this equation. However, it can be solved using trigonometric identities and techniques.

What is the range of values for which the equation has solutions?

The equation has solutions for all real values of x.

Can the equation be solved algebraically?

No, the equation cannot be solved algebraically. It requires the use of trigonometric identities and techniques.

Can the equation have more than one solution for a given value of x?

Yes, the equation can have multiple solutions for a given value of x. This is because trigonometric functions are periodic and have multiple solutions within a given interval.

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