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Popular Trigonometry >

arctan(100x)-arctan(x)=45

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Solution

arctan(100x)−arctan(x)=45∘

Solution

x=20099+9401​​,x=20099−9401​​
Solution steps
arctan(100x)−arctan(x)=45∘
Rewrite using trig identities
arctan(100x)−arctan(x)
Use the Sum to Product identity: arctan(s)−arctan(t)=arctan(1+sts−t​)=arctan(1+100xx100x−x​)
arctan(1+100xx100x−x​)=45∘
Apply trig inverse properties
arctan(1+100xx100x−x​)=45∘
arctan(x)=a⇒x=tan(a)1+100xx100x−x​=tan(45∘)
tan(45∘)=1
tan(45∘)
Use the following trivial identity:tan(45∘)=1
tan(45∘)
tan(x) periodicity table with 180∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​tan(x)033​​13​±∞−3​−1−33​​​​
=1
=1
1+100xx100x−x​=1
1+100xx100x−x​=1
Solve 1+100xx100x−x​=1:x=20099+9401​​,x=20099−9401​​
1+100xx100x−x​=1
Simplify 1+100xx100x−x​:1+100x299x​
1+100xx100x−x​
Add similar elements: 100x−x=99x=1+100xx99x​
1+100xx=1+100x2
1+100xx
100xx=100x2
100xx
Apply exponent rule: ab⋅ac=ab+cxx=x1+1=100x1+1
Add the numbers: 1+1=2=100x2
=1+100x2
=1+100x299x​
1+100x299x​=1
Multiply both sides by 1+100x2
1+100x299x​=1
Multiply both sides by 1+100x21+100x299x​(1+100x2)=1⋅(1+100x2)
Simplify
1+100x299x​(1+100x2)=1⋅(1+100x2)
Simplify 1+100x299x​(1+100x2):99x
1+100x299x​(1+100x2)
Multiply fractions: a⋅cb​=ca⋅b​=1+100x299x(1+100x2)​
Cancel the common factor: 1+100x2=99x
Simplify 1⋅(1+100x2):1+100x2
1⋅(1+100x2)
Multiply: 1⋅(1+100x2)=(1+100x2)=(1+100x2)
Remove parentheses: (a)=a=1+100x2
99x=1+100x2
99x=1+100x2
99x=1+100x2
Solve 99x=1+100x2:x=20099+9401​​,x=20099−9401​​
99x=1+100x2
Switch sides1+100x2=99x
Move 99xto the left side
1+100x2=99x
Subtract 99x from both sides1+100x2−99x=99x−99x
Simplify1+100x2−99x=0
1+100x2−99x=0
Write in the standard form ax2+bx+c=0100x2−99x+1=0
Solve with the quadratic formula
100x2−99x+1=0
Quadratic Equation Formula:
For a=100,b=−99,c=1x1,2​=2⋅100−(−99)±(−99)2−4⋅100⋅1​​
x1,2​=2⋅100−(−99)±(−99)2−4⋅100⋅1​​
(−99)2−4⋅100⋅1​=9401​
(−99)2−4⋅100⋅1​
Apply exponent rule: (−a)n=an,if n is even(−99)2=992=992−4⋅100⋅1​
Multiply the numbers: 4⋅100⋅1=400=992−400​
992=9801=9801−400​
Subtract the numbers: 9801−400=9401=9401​
x1,2​=2⋅100−(−99)±9401​​
Separate the solutionsx1​=2⋅100−(−99)+9401​​,x2​=2⋅100−(−99)−9401​​
x=2⋅100−(−99)+9401​​:20099+9401​​
2⋅100−(−99)+9401​​
Apply rule −(−a)=a=2⋅10099+9401​​
Multiply the numbers: 2⋅100=200=20099+9401​​
x=2⋅100−(−99)−9401​​:20099−9401​​
2⋅100−(−99)−9401​​
Apply rule −(−a)=a=2⋅10099−9401​​
Multiply the numbers: 2⋅100=200=20099−9401​​
The solutions to the quadratic equation are:x=20099+9401​​,x=20099−9401​​
x=20099+9401​​,x=20099−9401​​
x=20099+9401​​,x=20099−9401​​
Verify solutions by plugging them into the original equation
Check the solutions by plugging them into arctan(100x)−arctan(x)=45∘
Remove the ones that don't agree with the equation.
Check the solution 20099+9401​​:True
20099+9401​​
Plug in n=120099+9401​​
For arctan(100x)−arctan(x)=45∘plug inx=20099+9401​​arctan(100⋅20099+9401​​)−arctan(20099+9401​​)=45∘
Refine0.78539…=0.78539…
⇒True
Check the solution 20099−9401​​:True
20099−9401​​
Plug in n=120099−9401​​
For arctan(100x)−arctan(x)=45∘plug inx=20099−9401​​arctan(100⋅20099−9401​​)−arctan(20099−9401​​)=45∘
Refine0.78539…=0.78539…
⇒True
x=20099+9401​​,x=20099−9401​​

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