# Solve for x

Find all values of x which satisfies this equation,

sin5x+cos3x=1

1057
Ketan Chandak ·

Since max value of sin5x+cos3x is 1 it'll be one only when either sin or cos x is 1.

1708
man111 singh ·

\hspace{-20}$Given$\bf{\sin^5 x+\cos^3 x = 1 = \sin^2 x+\cos^2 x}$\\\\\\$\bf{\Rightarrow \sin^5 x-\sin^2 x+\cos^3 x-\cos^2 x = 1}$\\\\\\$\bf{\Rightarrow \sin^2 x \cdot \left(\sin^3 x-1\right)+\cos^2 x\cdot (\cos x - 1) = 0}$\\\\\\ Now Using$\bf{\bullet -1\leq \sin x\;, \cos x \leq 1 }$\\\\\\ So we get$\bf{\sin^2 x\cdot \left(\sin^3 x-1\right)\leq 0}$and$\bf{\cos^2 x\cdot (\cos x-1)\leq 0}$\\\\\\ Now equation is satisfied when$\bf{\sin^2 x\cdot \left(\sin^3 x-1\right)\;\;,\cos^2 x\cdot \left(\cos x-1\right)=0}$\\\\\\ So$\bf{\sin^2 x\cdot \left(\sin^3 x-1\right)=0\;\; \cap \;\; \cos^2 x\cdot \left(\cos x-1\right)=0}$\\\\\\ Where$\bf{\cap}$means Intersection.....\\\\\\$\bf{\bullet \;\; }$If$\bf{\sin^2 x\cdot (\sin^3 x-1) = 0\;,}$Then either$\bf{\sin^2 x= 0\Rightarrow x=n\pi}$\\\\\\ Or$\bf{\sin^3 x -1 = 0\Rightarrow \sin x -1=0\Rightarrow \sin x= 1\Rightarrow x = 2n\pi+\frac{\pi}{2}}$\\\\\\ Where$\bf{n\in \mathbb{Z}.}$\\\\\\ \hspace{-20}\bf{\bullet \;\; }$ If $\bf{\cos^2 x\cdot (\cos x-1) = 0\;,}$ Then either $\bf{\cos^2 x= 0\Rightarrow x=2n\pi\pm \frac{\pi}{2}}$\\\\\\ Or $\bf{\cos x-1 = 0\Rightarrow \cos x=1\Rightarrow x=2n\pi}$\\\\\\ Where $\bf{n\in \mathbb{Z}.}$\\\\\\ So from These $\bf{2}$ cases, we get $\bf{x = 2k\pi}$ and $\bf{x=2k\pi+\frac{\pi}{2}}$\\\\\\ Where $\bf{k\in \mathbb{Z}.}$

49
Pratyasha Das ·

x can also be equal to -1/2sin^-1(2/5)

• Manish Shankar How? Show me your method
1357
Manish Shankar ·

anyone for this

3
Shambo Mohanty ·

sin5Î±+cos3Î±=1
then we can say that,sin5Î±â‰¤sin2Î± and cos3Î±â‰¤cos2Î±
here the case is in equality,then sin5Î±=sin2Î± and cos3Î±=cos2Î±
hence there are 2 cases
1> when sin2Î±=0 and cos2Î±=1
2> when sin2Î±=1 and cos2Î±=0
in 1st case the general solution is 2nÏ€
in 2nd case the general solution is 2kÏ€+Ï€/2
[n belongs to I]