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Popular Trigonometry >

tan(θ)-3cot(θ)=0

  • Pre Algebra
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Solution

tan(θ)−3cot(θ)=0

Solution

θ=1.04719…+πn,θ=2.09439…+πn
+1
Degrees
θ=60∘+180∘n,θ=120∘+180∘n
Solution steps
tan(θ)−3cot(θ)=0
Rewrite using trig identities
tan(θ)−3cot(θ)
Use the basic trigonometric identity: tan(x)=cot(x)1​=cot(θ)1​−3cot(θ)
cot(θ)1​−3cot(θ)=0
Solve by substitution
cot(θ)1​−3cot(θ)=0
Let: cot(θ)=uu1​−3u=0
u1​−3u=0:u=31​​,u=−31​​
u1​−3u=0
Multiply both sides by u
u1​−3u=0
Multiply both sides by uu1​u−3uu=0⋅u
Simplify
u1​u−3uu=0⋅u
Simplify u1​u:1
u1​u
Multiply fractions: a⋅cb​=ca⋅b​=u1⋅u​
Cancel the common factor: u=1
Simplify −3uu:−3u2
−3uu
Apply exponent rule: ab⋅ac=ab+cuu=u1+1=−3u1+1
Add the numbers: 1+1=2=−3u2
Simplify 0⋅u:0
0⋅u
Apply rule 0⋅a=0=0
1−3u2=0
1−3u2=0
1−3u2=0
Solve 1−3u2=0:u=31​​,u=−31​​
1−3u2=0
Move 1to the right side
1−3u2=0
Subtract 1 from both sides1−3u2−1=0−1
Simplify−3u2=−1
−3u2=−1
Divide both sides by −3
−3u2=−1
Divide both sides by −3−3−3u2​=−3−1​
Simplifyu2=31​
u2=31​
For x2=f(a) the solutions are x=f(a)​,−f(a)​
u=31​​,u=−31​​
u=31​​,u=−31​​
Verify Solutions
Find undefined (singularity) points:u=0
Take the denominator(s) of u1​−3u and compare to zero
u=0
The following points are undefinedu=0
Combine undefined points with solutions:
u=31​​,u=−31​​
Substitute back u=cot(θ)cot(θ)=31​​,cot(θ)=−31​​
cot(θ)=31​​,cot(θ)=−31​​
cot(θ)=31​​:θ=arccot(31​​)+πn
cot(θ)=31​​
Apply trig inverse properties
cot(θ)=31​​
General solutions for cot(θ)=31​​cot(x)=a⇒x=arccot(a)+πnθ=arccot(31​​)+πn
θ=arccot(31​​)+πn
cot(θ)=−31​​:θ=arccot(−31​​)+πn
cot(θ)=−31​​
Apply trig inverse properties
cot(θ)=−31​​
General solutions for cot(θ)=−31​​cot(x)=−a⇒x=arccot(−a)+πnθ=arccot(−31​​)+πn
θ=arccot(−31​​)+πn
Combine all the solutionsθ=arccot(31​​)+πn,θ=arccot(−31​​)+πn
Show solutions in decimal formθ=1.04719…+πn,θ=2.09439…+πn

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Popular Examples

cot(-pi/2)sin^2(x)+sin(x)=0sin(15)sec(120)csc(90)

Frequently Asked Questions (FAQ)

  • What is the general solution for tan(θ)-3cot(θ)=0 ?

    The general solution for tan(θ)-3cot(θ)=0 is θ=1.04719…+pin,θ=2.09439…+pin
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