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

The general solution for 3tan(x)=cot(x) is x= pi/6+pin,x=(5pi)/6+pin

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3tan(x)=cot(x)

Solution

3 tan (x) = cot (x)

Solution

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

3 tan (x) = cot (x)

Subtract

cot (x)

from both sides

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

Rewrite using trig identities

- cot (x) + 3 tan (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

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

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

Solve by substitution

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

Let:

cot (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

- uu : - u^{2}

- uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

Simplify

\frac{3}{u} u : 3

\frac{3}{u} u

Multiply fractions:

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

= \frac{3 u}{u}

Cancel the common factor:

u

= 3

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- u^{2} + 3 = 0

- u^{2} + 3 = 0

- u^{2} + 3 = 0

Solve

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

- u^{2} + 3 = 0

Move

3

to the right side

- u^{2} + 3 = 0

Subtract

3

from both sides

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

Simplify

- u^{2} = - 3

- u^{2} = - 3

Divide both sides by

- 1

- u^{2} = - 3

Divide both sides by

- 1

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

Simplify

u^{2} = 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

u = 3, u = - 3

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- u + \frac{3}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3, u = - 3

Substitute back

u = cot (x)

cot (x) = 3, cot (x) = - 3

cot (x) = 3, cot (x) = - 3

cot (x) = 3 : x = \frac{π}{6} + πn

cot (x) = 3

General solutions for

cot (x) = 3

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

cot (x) = - 3 : x = \frac{5 π}{6} + πn

cot (x) = - 3

General solutions for

cot (x) = - 3

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

Combine all the solutions

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

Popular Examples

cos(θ)= 7/25

cos (θ) = \frac{7}{25}

2tan^2(θ)sin(θ)-tan^2(θ)=0

2 tan^{2} (θ) sin (θ) - tan^{2} (θ) = 0

sin^2(x)=-3sin(x)+4

sin^{2} (x) = - 3 sin (x) + 4

tan^2(x)csc(x)=tan^2(x)

tan^{2} (x) csc (x) = tan^{2} (x)

12.3=2.3sin(24x)+14.1

12.3 = 2.3 sin (24 x) + 14.1

Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x)=cot(x) ?
The general solution for 3tan(x)=cot(x) is x= pi/6+pin,x=(5pi)/6+pin