What is the general solution for 2tan(x)-3cot(x)=0 ?

The general solution for 2tan(x)-3cot(x)=0 is x=0.88607…+pin,x=2.25551…+pin

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2tan(x)-3cot(x)=0

Solution

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

Solution

x = 0.88607 \dots + πn, x = 2.25551 \dots + πn

Degrees

x = 50.76847 \dots^{\circ} + 18 0^{\circ} n, x = 129.23152 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

2 tan (x) - 3 cot (x)

Use the basic trigonometric identity:

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

= 2 \cdot \frac{1}{cot ( x )} - 3 cot (x)

2 \cdot \frac{1}{cot ( x )} = \frac{2}{cot ( x )}

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

Multiply fractions:

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

= \frac{1 \cdot 2}{cot ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{cot ( x )}

= \frac{2}{cot ( x )} - 3 cot (x)

\frac{2}{cot ( x )} - 3 cot (x) = 0

Solve by substitution

\frac{2}{cot ( x )} - 3 cot (x) = 0

Let:

cot (x) = u

\frac{2}{u} - 3 u = 0

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

\frac{2}{u} - 3 u = 0

Multiply both sides by

u

\frac{2}{u} - 3 u = 0

Multiply both sides by

u

\frac{2}{u} u - 3 uu = 0 \cdot u

Simplify

\frac{2}{u} u - 3 uu = 0 \cdot u

Simplify

\frac{2}{u} u : 2

\frac{2}{u} u

Multiply fractions:

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

= \frac{2 u}{u}

Cancel the common factor:

u

= 2

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

2 - 3 u^{2} = 0

2 - 3 u^{2} = 0

2 - 3 u^{2} = 0

Solve

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

2 - 3 u^{2} = 0

Move

2

to the right side

2 - 3 u^{2} = 0

Subtract

2

from both sides

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

Simplify

- 3 u^{2} = - 2

- 3 u^{2} = - 2

Divide both sides by

- 3

- 3 u^{2} = - 2

Divide both sides by

- 3

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{2}{3}, u = - \frac{2}{3}

u = \frac{2}{3}, u = - \frac{2}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{2}{u} - 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{2}{3}, u = - \frac{2}{3}

Substitute back

u = cot (x)

cot (x) = \frac{2}{3}, cot (x) = - \frac{2}{3}

cot (x) = \frac{2}{3}, cot (x) = - \frac{2}{3}

cot (x) = \frac{2}{3} : x = arccot (\frac{2}{3}) + πn

cot (x) = \frac{2}{3}

Apply trig inverse properties

cot (x) = \frac{2}{3}

General solutions for

cot (x) = \frac{2}{3}

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

x = arccot (\frac{2}{3}) + πn

x = arccot (\frac{2}{3}) + πn

cot (x) = - \frac{2}{3} : x = arccot (- \frac{2}{3}) + πn

cot (x) = - \frac{2}{3}

Apply trig inverse properties

cot (x) = - \frac{2}{3}

General solutions for

cot (x) = - \frac{2}{3}

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

x = arccot (- \frac{2}{3}) + πn

x = arccot (- \frac{2}{3}) + πn

Combine all the solutions

x = arccot (\frac{2}{3}) + πn, x = arccot (- \frac{2}{3}) + πn

Show solutions in decimal form

x = 0.88607 \dots + πn, x = 2.25551 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2tan(x)-3cot(x)=0 ?
The general solution for 2tan(x)-3cot(x)=0 is x=0.88607…+pin,x=2.25551…+pin