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Popular Trigonometry >

tan(x)+sqrt(3)=1+sqrt(3)cot(x)

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Solution

tan(x)+3​=1+3​cot(x)

Solution

x=32π​+πn,x=4π​+πn
+1
Degrees
x=120∘+180∘n,x=45∘+180∘n
Solution steps
tan(x)+3​=1+3​cot(x)
Subtract 1+3​cot(x) from both sidestan(x)+3​−1−3​cot(x)=0
Rewrite using trig identities
−1+3​+tan(x)−cot(x)3​
Use the basic trigonometric identity: tan(x)=cot(x)1​=−1+3​+cot(x)1​−cot(x)3​
−1+cot(x)1​+3​−cot(x)3​=0
Solve by substitution
−1+cot(x)1​+3​−cot(x)3​=0
Let: cot(x)=u−1+u1​+3​−u3​=0
−1+u1​+3​−u3​=0:u=−33​​,u=1
−1+u1​+3​−u3​=0
Multiply both sides by u
−1+u1​+3​−u3​=0
Multiply both sides by u−1⋅u+u1​u+3​u−u3​u=0⋅u
Simplify
−1⋅u+u1​u+3​u−u3​u=0⋅u
Simplify −1⋅u:−u
−1⋅u
Multiply: 1⋅u=u=−u
Simplify u1​u:1
u1​u
Multiply fractions: a⋅cb​=ca⋅b​=u1⋅u​
Cancel the common factor: u=1
Simplify −u3​u:−3​u2
−u3​u
Apply exponent rule: ab⋅ac=ab+cuu=u1+1=−3​u1+1
Add the numbers: 1+1=2=−3​u2
Simplify 0⋅u:0
0⋅u
Apply rule 0⋅a=0=0
−u+1+3​u−3​u2=0
−u+1+3​u−3​u2=0
−u+1+3​u−3​u2=0
Solve −u+1+3​u−3​u2=0:u=−33​​,u=1
−u+1+3​u−3​u2=0
Write in the standard form ax2+bx+c=0−3​u2+(−1+3​)u+1=0
Solve with the quadratic formula
−3​u2+(−1+3​)u+1=0
Quadratic Equation Formula:
For a=−3​,b=−1+3​,c=1u1,2​=2(−3​)−(−1+3​)±(−1+3​)2−4(−3​)⋅1​​
u1,2​=2(−3​)−(−1+3​)±(−1+3​)2−4(−3​)⋅1​​
(−1+3​)2−4(−3​)⋅1​=3​+1
(−1+3​)2−4(−3​)⋅1​
Apply rule −(−a)=a=(−1+3​)2+43​⋅1​
Multiply the numbers: 4⋅1=4=(3​−1)2+43​​
Expand (−1+3​)2+43​:4+23​
(−1+3​)2+43​
(−1+3​)2:4−23​
Apply Perfect Square Formula: (a+b)2=a2+2ab+b2a=−1,b=3​
=(−1)2+2(−1)3​+(3​)2
Simplify (−1)2+2(−1)3​+(3​)2:4−23​
(−1)2+2(−1)3​+(3​)2
Remove parentheses: (−a)=−a=(−1)2−2⋅1⋅3​+(3​)2
(−1)2=1
(−1)2
Apply exponent rule: (−a)n=an,if n is even(−1)2=12=12
Apply rule 1a=1=1
2⋅1⋅3​=23​
2⋅1⋅3​
Multiply the numbers: 2⋅1=2=23​
(3​)2=3
(3​)2
Apply radical rule: a​=a21​=(321​)2
Apply exponent rule: (ab)c=abc=321​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3
=1−23​+3
Add the numbers: 1+3=4=4−23​
=4−23​
=4−23​+43​
Add similar elements: −23​+43​=23​=4+23​
=4+23​​
=3+23​+1​
=(3​)2+23​+(1​)2​
1​=1
1​
Apply rule 1​=1=1
=(3​)2+23​+12​
23​⋅1=23​
23​⋅1
Multiply the numbers: 2⋅1=2=23​
=(3​)2+23​⋅1+12​
Apply Perfect Square Formula: (a+b)2=a2+2ab+b2(3​)2+23​⋅1+12=(3​+1)2=(3​+1)2​
Apply radical rule: (3​+1)2​=3​+1=3​+1
u1,2​=2(−3​)−(−1+3​)±(3​+1)​
Separate the solutionsu1​=2(−3​)−(−1+3​)+3​+1​,u2​=2(−3​)−(−1+3​)−(3​+1)​
u=2(−3​)−(−1+3​)+3​+1​:−33​​
2(−3​)−(−1+3​)+3​+1​
Remove parentheses: (−a)=−a=−23​−(−1+3​)+3​+1​
Apply the fraction rule: −ba​=−ba​=−23​−(−1+3​)+3​+1​
Expand −(−1+3​)+3​+1:2
−(−1+3​)+3​+1
−(−1+3​):1−3​
−(−1+3​)
Distribute parentheses=−(−1)−(3​)
Apply minus-plus rules−(−a)=a,−(a)=−a=1−3​
=1−3​+3​+1
Simplify 1−3​+3​+1:2
1−3​+3​+1
Add similar elements: −3​+3​=0=1+1
Add the numbers: 1+1=2=2
=2
=−23​2​
Divide the numbers: 22​=1=−3​1​
Rationalize −3​1​:−33​​
−3​1​
Multiply by the conjugate 3​3​​=−3​3​1⋅3​​
1⋅3​=3​
3​3​=3
3​3​
Apply radical rule: a​a​=a3​3​=3=3
=−33​​
=−33​​
u=2(−3​)−(−1+3​)−(3​+1)​:1
2(−3​)−(−1+3​)−(3​+1)​
Remove parentheses: (−a)=−a=−23​−(−1+3​)−(3​+1)​
Apply the fraction rule: −b−a​=ba​−(−1+3​)−(3​+1)=−((1+3​)+(3​−1))=23​(1+3​)+(3​−1)​
Remove parentheses: (a)=a=23​1+3​+3​−1​
1+3​+3​−1=23​
1+3​+3​−1
Add similar elements: 3​+3​=23​=1+23​−1
1−1=0=23​
=23​23​​
Apply rule aa​=1=1
The solutions to the quadratic equation are:u=−33​​,u=1
u=−33​​,u=1
Verify Solutions
Find undefined (singularity) points:u=0
Take the denominator(s) of −1+u1​+3​−u3​ and compare to zero
u=0
The following points are undefinedu=0
Combine undefined points with solutions:
u=−33​​,u=1
Substitute back u=cot(x)cot(x)=−33​​,cot(x)=1
cot(x)=−33​​,cot(x)=1
cot(x)=−33​​:x=32π​+πn
cot(x)=−33​​
General solutions for cot(x)=−33​​
cot(x) periodicity table with πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​cot(x)∓∞3​133​​0−33​​−1−3​​​
x=32π​+πn
x=32π​+πn
cot(x)=1:x=4π​+πn
cot(x)=1
General solutions for cot(x)=1
cot(x) periodicity table with πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​cot(x)∓∞3​133​​0−33​​−1−3​​​
x=4π​+πn
x=4π​+πn
Combine all the solutionsx=32π​+πn,x=4π​+πn

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Popular Examples

tan(u)-cot(u)=2sin(x)sin(x)=015=arctan((0.375)/(2x))cos(θ)(1+cos(θ))=sin^2(θ)cos(θ)=-0.8,sin(pi^2-θ)

Frequently Asked Questions (FAQ)

  • What is the general solution for tan(x)+sqrt(3)=1+sqrt(3)cot(x) ?

    The general solution for tan(x)+sqrt(3)=1+sqrt(3)cot(x) is x=(2pi)/3+pin,x= pi/4+pin
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