What is the general solution for 3cot(x)=tan(x) ?

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

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Popular Trigonometry >

3cot(x)=tan(x)

Solution

3 cot (x) = tan (x)

Solution

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

3 cot (x) = tan (x)

Subtract

tan (x)

from both sides

3 cot (x) - tan (x) = 0

Rewrite using trig identities

- tan (x) + 3 cot (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= - \frac{1}{cot ( x )} + 3 cot (x)

- \frac{1}{cot ( x )} + 3 cot (x) = 0

Solve by substitution

- \frac{1}{cot ( x )} + 3 cot (x) = 0

Let:

cot (x) = u

- \frac{1}{u} + 3 u = 0

- \frac{1}{u} + 3 u = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

- \frac{1}{u} + 3 u = 0

Multiply both sides by

u

- \frac{1}{u} + 3 u = 0

Multiply both sides by

u

- \frac{1}{u} u + 3 uu = 0 \cdot u

Simplify

- \frac{1}{u} u + 3 uu = 0 \cdot u

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

3 uu : 3 u^{2}

3 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 1 + 3 u^{2} = 0

- 1 + 3 u^{2} = 0

- 1 + 3 u^{2} = 0

Solve

- 1 + 3 u^{2} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

- 1 + 3 u^{2} = 0

Move

1

to the right side

- 1 + 3 u^{2} = 0

Add

1

to both sides

- 1 + 3 u^{2} + 1 = 0 + 1

Simplify

3 u^{2} = 1

3 u^{2} = 1

Divide both sides by

3

3 u^{2} = 1

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{1}{3}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

u = \frac{1}{3}, u = - \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- \frac{1}{u} + 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = cot (x)

cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = \frac{1}{3} : x = arccot (\frac{1}{3}) + πn

cot (x) = \frac{1}{3}

Apply trig inverse properties

cot (x) = \frac{1}{3}

General solutions for

cot (x) = \frac{1}{3}

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (\frac{1}{3}) + πn

x = arccot (\frac{1}{3}) + πn

cot (x) = - \frac{1}{3} : x = arccot (- \frac{1}{3}) + πn

cot (x) = - \frac{1}{3}

Apply trig inverse properties

cot (x) = - \frac{1}{3}

General solutions for

cot (x) = - \frac{1}{3}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{1}{3}) + πn

x = arccot (- \frac{1}{3}) + πn

Combine all the solutions

x = arccot (\frac{1}{3}) + πn, x = arccot (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

Graph

Popular Examples

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2sin^2(x)-4sin(x)-6=0

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2sin(A)-sqrt(3)=0

Frequently Asked Questions (FAQ)

What is the general solution for 3cot(x)=tan(x) ?
The general solution for 3cot(x)=tan(x) is x=1.04719…+pin,x=2.09439…+pin