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Popular Trigonometry >

solvefor u,x=4sinh(u)

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Solution

solvefor

Solution

u=ln(4x+x2+16​​),u=ln(4x−x2+16​​)
Solution steps
x=4sinh(u)
Switch sides4sinh(u)=x
Rewrite using trig identities
4sinh(u)=x
Use the Hyperbolic identity: sinh(x)=2ex−e−x​4⋅2eu−e−u​=x
4⋅2eu−e−u​=x
4⋅2eu−e−u​=x:u=ln(4x+x2+16​​),u=ln(4x−x2+16​​)
4⋅2eu−e−u​=x
Apply exponent rules
4⋅2eu−e−u​=x
Apply exponent rule: abc=(ab)ce−u=(eu)−14⋅2eu−(eu)−1​=x
4⋅2eu−(eu)−1​=x
Rewrite the equation with eu=v4⋅2v−v−1​=x
Solve 4⋅2v−v−1​=x:v=4x+x2+16​​,v=4x−x2+16​​
4⋅2v−v−1​=x
Refinev2(v2−1)​=x
Multiply both sides by v
v2(v2−1)​=x
Multiply both sides by vv2(v2−1)​v=xv
Simplify2(v2−1)=xv
2(v2−1)=xv
Solve 2(v2−1)=xv:v=4x+x2+16​​,v=4x−x2+16​​
2(v2−1)=xv
Expand 2(v2−1):2v2−2
2(v2−1)
Apply the distributive law: a(b−c)=ab−aca=2,b=v2,c=1=2v2−2⋅1
Multiply the numbers: 2⋅1=2=2v2−2
2v2−2=xv
Move xvto the left side
2v2−2=xv
Subtract xv from both sides2v2−2−xv=xv−xv
Simplify2v2−2−xv=0
2v2−2−xv=0
Write in the standard form ax2+bx+c=02v2−xv−2=0
Solve with the quadratic formula
2v2−xv−2=0
Quadratic Equation Formula:
For a=2,b=−x,c=−2v1,2​=2⋅2−(−x)±(−x)2−4⋅2(−2)​​
v1,2​=2⋅2−(−x)±(−x)2−4⋅2(−2)​​
Simplify (−x)2−4⋅2(−2)​:x2+16​
(−x)2−4⋅2(−2)​
Apply rule −(−a)=a=(−x)2+4⋅2⋅2​
Apply exponent rule: (−a)n=an,if n is even(−x)2=x2=x2+4⋅2⋅2​
Multiply the numbers: 4⋅2⋅2=16=x2+16​
v1,2​=2⋅2−(−x)±x2+16​​
Separate the solutionsv1​=2⋅2−(−x)+x2+16​​,v2​=2⋅2−(−x)−x2+16​​
v=2⋅2−(−x)+x2+16​​:4x+x2+16​​
2⋅2−(−x)+x2+16​​
Apply rule −(−a)=a=2⋅2x+x2+16​​
Multiply the numbers: 2⋅2=4=4x+x2+16​​
v=2⋅2−(−x)−x2+16​​:4x−x2+16​​
2⋅2−(−x)−x2+16​​
Apply rule −(−a)=a=2⋅2x−x2+16​​
Multiply the numbers: 2⋅2=4=4x−x2+16​​
The solutions to the quadratic equation are:v=4x+x2+16​​,v=4x−x2+16​​
v=4x+x2+16​​,v=4x−x2+16​​
v=4x+x2+16​​,v=4x−x2+16​​
Substitute back v=eu,solve for u
Solve eu=4x+x2+16​​:u=ln(4x+x2+16​​)
eu=4x+x2+16​​
Apply exponent rules
eu=4x+x2+16​​
If f(x)=g(x), then ln(f(x))=ln(g(x))ln(eu)=ln(4x+x2+16​​)
Apply log rule: ln(ea)=aln(eu)=uu=ln(4x+x2+16​​)
u=ln(4x+x2+16​​)
Solve eu=4x−x2+16​​:u=ln(4x−x2+16​​)
eu=4x−x2+16​​
Apply exponent rules
eu=4x−x2+16​​
If f(x)=g(x), then ln(f(x))=ln(g(x))ln(eu)=ln(4x−x2+16​​)
Apply log rule: ln(ea)=aln(eu)=uu=ln(4x−x2+16​​)
u=ln(4x−x2+16​​)
u=ln(4x+x2+16​​),u=ln(4x−x2+16​​)
u=ln(4x+x2+16​​),u=ln(4x−x2+16​​)

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tan(x)= 25/32sin(x+20)+sin(40-x)=1solvefor x,f=2cos(3x^2-1)entoncesftan(a)= 5/8solvefor x,z=tan(x/2)

Frequently Asked Questions (FAQ)

  • What is the general solution for solvefor u,x=4sinh(u) ?

    The general solution for solvefor u,x=4sinh(u) is u=ln((x+sqrt(x^2+16))/4),u=ln((x-sqrt(x^2+16))/4)
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