Real Solutions for Sin^5 x + Cos^3 x = 1 | POTW #500 - April 29th 2022

  • MHB
  • Thread starter anemone
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In summary, the equation "Sin^5 x + Cos^3 x = 1" is a trigonometric equation that represents a relationship between the sine and cosine functions and has real-life applications in fields such as physics, engineering, and mathematics. "POTW #500 - April 29th 2022" stands for "Problem of the Week #500" and refers to the 500th problem that will be posted on April 29th, 2022, in a recurring event where a new problem is posted each week for people to solve and discuss. This equation can be solved using various techniques such as manipulating the equation algebraically, graphing, or using numerical methods, and can have multiple solutions due to the
  • #1
anemone
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MHB
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Here is this week's POTW:

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Find all the real solutions of the equation $\sin^5 x+\cos^3 x=1$.

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  • #2
Congratulations to the following members for their correct solution!(Cool)

1. Opalg
2. kaliprasad
3. lfdahl

Solution from Opalg:
Let $f(x) = \sin^5x+ \cos^3x - 1$. Since $f$ has period $2\pi$ it will be enough to find solutions of $f(x)=0$ in the interval $0\leqslant x < 2\pi$.

If $\sin x$ or $\cos x$ is negative then $f(x)<0$. That rules out the interval $\pi/2 <x < 2\pi$. So it will be enough to find solutions of $f(x)=0$ in the interval $0\leqslant x \leqslant \pi/2$.

If $x=0$ or $x=\pi/2$ then $f(x)=0$. That leaves the interval $0<x<\pi/2$. But in that interval $0<\sin x <1$ and $0<\cos x <1$. Therefore $ \sin^5x+ \cos^3x < \sin^2x+ \cos^2x =1$ and so $f(x) <0$.

Thus the only solutions are $x = 2k\pi$ and $x = \left(2k+\frac12\right)\pi$ for $k\in\Bbb{Z}$.
 

1. What is the purpose of solving the equation Sin^5 x + Cos^3 x = 1?

The purpose of solving this equation is to find the values of x that satisfy the equation and provide a real solution. This can help in understanding the behavior and properties of trigonometric functions.

2. What are the steps involved in solving this equation?

The steps involved in solving this equation include simplifying the equation using trigonometric identities, isolating the trigonometric function with the highest power, and using inverse trigonometric functions to find the values of x.

3. Can this equation have more than one real solution?

Yes, this equation can have more than one real solution. In fact, it can have an infinite number of solutions since trigonometric functions are periodic.

4. How can I check if my solution is correct?

You can check if your solution is correct by substituting the value of x into the original equation and seeing if it satisfies the equation. You can also use a graphing calculator to plot the equation and see if the x-values of the intersection points are the same as your solution.

5. What are the practical applications of solving this equation?

Solving this equation can have practical applications in fields such as engineering, physics, and astronomy. It can also be used to model and analyze periodic phenomena in nature, such as the motion of waves and vibrations.

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