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Popular Trigonometry >

solvefor x,f=2cos(3x^2-1)entoncesf

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Solution

solvefor

Solution

x=3arccos(2e2cn2tos1​)+2πk+1​​,x=−3arccos(2e2cn2tos1​)+2πk+1​​,x=3−arccos(2e2cn2tos1​)+2πk+1​​,x=−3−arccos(2e2cn2tos1​)+2πk+1​​
Solution steps
f=2cos(3x2−1)entoncesf
Switch sides2cos(3x2−1)entoncesf=f
Divide both sides by 2entoncesf;c=0
2cos(3x2−1)entoncesf=f
Divide both sides by 2entoncesf;c=02entoncesf2cos(3x2−1)entoncesf​=2entoncesff​;c=0
Simplify
2entoncesf2cos(3x2−1)entoncesf​=2entoncesff​
Simplify 2entoncesf2cos(3x2−1)entoncesf​:cos(3x2−1)
2entoncesf2cos(3x2−1)entoncesf​
2cos(3x2−1)entoncesf=2e2cftosn2cos(3x2−1)
2cos(3x2−1)entoncesf
Apply exponent rule: ab⋅ac=ab+cee=e1+1=2cos(3x2−1)ntonce1+1sf
Add the numbers: 1+1=2=2cos(3x2−1)ntonce2sf
Apply exponent rule: ab⋅ac=ab+cnn=n1+1=2cos(3x2−1)ton1+1ce2sf
Add the numbers: 1+1=2=2cos(3x2−1)ton2ce2sf
=2eecftosnn2e2cftosn2cos(3x2−1)​
2entoncesf=2e2cftosn2
2entoncesf
Apply exponent rule: ab⋅ac=ab+cee=e1+1=2ntonce1+1sf
Add the numbers: 1+1=2=2ntonce2sf
Apply exponent rule: ab⋅ac=ab+cnn=n1+1=2ton1+1ce2sf
Add the numbers: 1+1=2=2ton2ce2sf
=2e2cftosn22e2cftosn2cos(3x2−1)​
Divide the numbers: 22​=1=e2cftosn2e2cftosn2cos(3x2−1)​
Cancel the common factor: t=e2cfosn2e2cfosn2cos(3x2−1)​
Cancel the common factor: o=e2cfsn2e2cfsn2cos(3x2−1)​
Cancel the common factor: n2=e2cfse2cfscos(3x2−1)​
Cancel the common factor: c=e2fse2fscos(3x2−1)​
Cancel the common factor: e2=fsfscos(3x2−1)​
Cancel the common factor: s=ffcos(3x2−1)​
Cancel the common factor: f=cos(3x2−1)
Simplify 2entoncesff​:2e2ctosn21​
2entoncesff​
2entoncesf=2e2cftosn2
2entoncesf
Apply exponent rule: ab⋅ac=ab+cee=e1+1=2ntonce1+1sf
Add the numbers: 1+1=2=2ntonce2sf
Apply exponent rule: ab⋅ac=ab+cnn=n1+1=2ton1+1ce2sf
Add the numbers: 1+1=2=2ton2ce2sf
=2e2cftosn2f​
Cancel the common factor: f=2e2ctosn21​
cos(3x2−1)=2e2ctosn21​;c=0
cos(3x2−1)=2e2ctosn21​;c=0
cos(3x2−1)=2e2ctosn21​;c=0
Apply trig inverse properties
cos(3x2−1)=2e2ctosn21​
General solutions for cos(3x2−1)=2e2ctosn21​cos(x)=a⇒x=arccos(a)+2πk,x=−arccos(a)+2πk3x2−1=arccos(2e2ctosn21​)+2πk,3x2−1=−arccos(2e2ctosn21​)+2πk
3x2−1=arccos(2e2ctosn21​)+2πk,3x2−1=−arccos(2e2ctosn21​)+2πk
Solve 3x2−1=arccos(2e2ctosn21​)+2πk:x=3arccos(2e2cn2tos1​)+2πk+1​​,x=−3arccos(2e2cn2tos1​)+2πk+1​​
3x2−1=arccos(2e2ctosn21​)+2πk
Move 1to the right side
3x2−1=arccos(2e2ctosn21​)+2πk
Add 1 to both sides3x2−1+1=arccos(2e2ctosn21​)+2πk+1
Simplify3x2=arccos(2e2ctosn21​)+2πk+1
3x2=arccos(2e2ctosn21​)+2πk+1
Divide both sides by 3
3x2=arccos(2e2ctosn21​)+2πk+1
Divide both sides by 333x2​=3arccos(2e2ctosn21​)​+32πk​+31​
Simplifyx2=3arccos(2e2ctosn21​)​+32πk​+31​
x2=3arccos(2e2ctosn21​)​+32πk​+31​
For x2=f(a) the solutions are x=f(a)​,−f(a)​
x=3arccos(2e2ctosn21​)​+32πk​+31​​,x=−3arccos(2e2ctosn21​)​+32πk​+31​​
Simplify 3arccos(2e2ctosn21​)​+32πk​+31​​:3arccos(2e2cn2tos1​)+2πk+1​​
3arccos(2e2ctosn21​)​+32πk​+31​​
Combine the fractions 3arccos(2e2cn2tos1​)​+32πk​+31​:3arccos(2e2cn2tos1​)+2πk+1​
Apply rule ca​±cb​=ca±b​=3arccos(2e2cn2tos1​)+2πk+1​
=3arccos(2e2ctosn21​)+2πk+1​​
=3arccos(2e2cn2tos1​)+2πk+1​​
Simplify −3arccos(2e2ctosn21​)​+32πk​+31​​:−3arccos(2e2cn2tos1​)+2πk+1​​
−3arccos(2e2ctosn21​)​+32πk​+31​​
Combine the fractions 3arccos(2e2cn2tos1​)​+32πk​+31​:3arccos(2e2cn2tos1​)+2πk+1​
Apply rule ca​±cb​=ca±b​=3arccos(2e2cn2tos1​)+2πk+1​
=−32πk+arccos(2e2cn2tos1​)+1​​
=−3arccos(2e2cn2tos1​)+2πk+1​​
x=3arccos(2e2cn2tos1​)+2πk+1​​,x=−3arccos(2e2cn2tos1​)+2πk+1​​
Solve 3x2−1=−arccos(2e2ctosn21​)+2πk:x=3−arccos(2e2cn2tos1​)+2πk+1​​,x=−3−arccos(2e2cn2tos1​)+2πk+1​​
3x2−1=−arccos(2e2ctosn21​)+2πk
Move 1to the right side
3x2−1=−arccos(2e2ctosn21​)+2πk
Add 1 to both sides3x2−1+1=−arccos(2e2ctosn21​)+2πk+1
Simplify3x2=−arccos(2e2ctosn21​)+2πk+1
3x2=−arccos(2e2ctosn21​)+2πk+1
Divide both sides by 3
3x2=−arccos(2e2ctosn21​)+2πk+1
Divide both sides by 333x2​=−3arccos(2e2ctosn21​)​+32πk​+31​
Simplifyx2=−3arccos(2e2ctosn21​)​+32πk​+31​
x2=−3arccos(2e2ctosn21​)​+32πk​+31​
For x2=f(a) the solutions are x=f(a)​,−f(a)​
x=−3arccos(2e2ctosn21​)​+32πk​+31​​,x=−−3arccos(2e2ctosn21​)​+32πk​+31​​
Simplify −3arccos(2e2ctosn21​)​+32πk​+31​​:3−arccos(2e2cn2tos1​)+2πk+1​​
−3arccos(2e2ctosn21​)​+32πk​+31​​
Combine the fractions −3arccos(2e2cn2tos1​)​+32πk​+31​:3−arccos(2e2cn2tos1​)+2πk+1​
Apply rule ca​±cb​=ca±b​=3−arccos(2e2cn2tos1​)+2πk+1​
=3−arccos(2e2ctosn21​)+2πk+1​​
=3−arccos(2e2cn2tos1​)+2πk+1​​
Simplify −−3arccos(2e2ctosn21​)​+32πk​+31​​:−3−arccos(2e2cn2tos1​)+2πk+1​​
−−3arccos(2e2ctosn21​)​+32πk​+31​​
Combine the fractions −3arccos(2e2cn2tos1​)​+32πk​+31​:3−arccos(2e2cn2tos1​)+2πk+1​
Apply rule ca​±cb​=ca±b​=3−arccos(2e2cn2tos1​)+2πk+1​
=−32πk+1−arccos(2e2cn2tos1​)​​
=−3−arccos(2e2cn2tos1​)+2πk+1​​
x=3−arccos(2e2cn2tos1​)+2πk+1​​,x=−3−arccos(2e2cn2tos1​)+2πk+1​​
x=3arccos(2e2cn2tos1​)+2πk+1​​,x=−3arccos(2e2cn2tos1​)+2πk+1​​,x=3−arccos(2e2cn2tos1​)+2πk+1​​,x=−3−arccos(2e2cn2tos1​)+2πk+1​​

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