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Popular Trigonometry >

solvefor y,sin(x^2y^2)=x

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Solution

solvefor

Solution

y=xarcsin(x)+2πn​​,y=−xarcsin(x)+2πn​​,y=xπ+arcsin(−x)+2πn​​,y=−xπ+arcsin(−x)+2πn​​
Solution steps
sin(x2y2)=x
Apply trig inverse properties
sin(x2y2)=x
General solutions for sin(x2y2)=xsin(x)=a⇒x=arcsin(a)+2πn,x=π+arcsin(a)+2πnx2y2=arcsin(x)+2πn,x2y2=π+arcsin(−x)+2πn
x2y2=arcsin(x)+2πn,x2y2=π+arcsin(−x)+2πn
Solve x2y2=arcsin(x)+2πn:y=xarcsin(x)+2πn​​,y=−xarcsin(x)+2πn​​;x=0
x2y2=arcsin(x)+2πn
Divide both sides by x2;x=0
x2y2=arcsin(x)+2πn
Divide both sides by x2;x=0x2x2y2​=x2arcsin(x)​+x22πn​;x=0
Simplifyy2=x2arcsin(x)​+x22πn​;x=0
y2=x2arcsin(x)​+x22πn​;x=0
For x2=f(a) the solutions are x=f(a)​,−f(a)​
y=x2arcsin(x)​+x22πn​​,y=−x2arcsin(x)​+x22πn​​;x=0
Simplify x2arcsin(x)​+x22πn​​:xarcsin(x)+2πn​​
x2arcsin(x)​+x22πn​​
Combine the fractions x2arcsin(x)​+x22πn​:x2arcsin(x)+2πn​
Apply rule ca​±cb​=ca±b​=x2arcsin(x)+2πn​
=x2arcsin(x)+2πn​​
Apply radical rule: assuming a≥0,b≥0=x2​arcsin(x)+2πn​​
Apply radical rule: assuming a≥0x2​=x=xarcsin(x)+2πn​​
Simplify −x2arcsin(x)​+x22πn​​:−xarcsin(x)+2πn​​
−x2arcsin(x)​+x22πn​​
Combine the fractions x2arcsin(x)​+x22πn​:x2arcsin(x)+2πn​
Apply rule ca​±cb​=ca±b​=x2arcsin(x)+2πn​
=−x22πn+arcsin(x)​​
Simplify x2arcsin(x)+2πn​​:xarcsin(x)+2πn​​
x2arcsin(x)+2πn​​
Apply radical rule: assuming a≥0,b≥0=x2​arcsin(x)+2πn​​
Apply radical rule: assuming a≥0x2​=x=xarcsin(x)+2πn​​
=−x2πn+arcsin(x)​​
=−xarcsin(x)+2πn​​
y=xarcsin(x)+2πn​​,y=−xarcsin(x)+2πn​​;x=0
Solve x2y2=π+arcsin(−x)+2πn:y=xπ+arcsin(−x)+2πn​​,y=−xπ+arcsin(−x)+2πn​​;x=0
x2y2=π+arcsin(−x)+2πn
Divide both sides by x2;x=0
x2y2=π+arcsin(−x)+2πn
Divide both sides by x2;x=0x2x2y2​=x2π​+x2arcsin(−x)​+x22πn​;x=0
Simplifyy2=x2π​+x2arcsin(−x)​+x22πn​;x=0
y2=x2π​+x2arcsin(−x)​+x22πn​;x=0
For x2=f(a) the solutions are x=f(a)​,−f(a)​
y=x2π​+x2arcsin(−x)​+x22πn​​,y=−x2π​+x2arcsin(−x)​+x22πn​​;x=0
Simplify x2π​+x2arcsin(−x)​+x22πn​​:xπ+arcsin(−x)+2πn​​
x2π​+x2arcsin(−x)​+x22πn​​
Combine the fractions x2π​+x2arcsin(−x)​+x22πn​:x2π+arcsin(−x)+2πn​
Apply rule ca​±cb​=ca±b​=x2π+arcsin(−x)+2πn​
=x2π+arcsin(−x)+2πn​​
Apply radical rule: assuming a≥0,b≥0=x2​π+arcsin(−x)+2πn​​
Apply radical rule: assuming a≥0x2​=x=xπ+arcsin(−x)+2πn​​
Simplify −x2π​+x2arcsin(−x)​+x22πn​​:−xπ+arcsin(−x)+2πn​​
−x2π​+x2arcsin(−x)​+x22πn​​
Combine the fractions x2π​+x2arcsin(−x)​+x22πn​:x2π+arcsin(−x)+2πn​
Apply rule ca​±cb​=ca±b​=x2π+arcsin(−x)+2πn​
=−x2π+2πn+arcsin(−x)​​
Simplify x2π+arcsin(−x)+2πn​​:xπ+arcsin(−x)+2πn​​
x2π+arcsin(−x)+2πn​​
Apply radical rule: assuming a≥0,b≥0=x2​π+arcsin(−x)+2πn​​
Apply radical rule: assuming a≥0x2​=x=xπ+arcsin(−x)+2πn​​
=−xπ+2πn+arcsin(−x)​​
=−xπ+arcsin(−x)+2πn​​
y=xπ+arcsin(−x)+2πn​​,y=−xπ+arcsin(−x)+2πn​​;x=0
y=xarcsin(x)+2πn​​,y=−xarcsin(x)+2πn​​,y=xπ+arcsin(−x)+2πn​​,y=−xπ+arcsin(−x)+2πn​​

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