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

The general solution for cot(x)-tan(x)=2sqrt(3) is x=0.26179…+pin,x=1.83259…+pin

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

Solution

cot (x) - tan (x) = 23

Solution

x = 0.26179 \dots + πn, x = 1.83259 \dots + πn

Degrees

x = 1 5^{\circ} + 18 0^{\circ} n, x = 10 5^{\circ} + 18 0^{\circ} n

Solution steps

cot (x) - tan (x) = 23

Subtract

23

from both sides

cot (x) - tan (x) - 23 = 0

Rewrite using trig identities

cot (x) - tan (x) - 23

Use the basic trigonometric identity:

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

= cot (x) - \frac{1}{cot ( x )} - 23

cot (x) - \frac{1}{cot ( x )} - 23 = 0

Solve by substitution

cot (x) - \frac{1}{cot ( x )} - 23 = 0

Let:

cot (x) = u

u - \frac{1}{u} - 23 = 0

u - \frac{1}{u} - 23 = 0 : u = 3 + 2, u = 3 - 2

u - \frac{1}{u} - 23 = 0

Multiply both sides by

u

u - \frac{1}{u} - 23 = 0

Multiply both sides by

u

uu - \frac{1}{u} u - 23 u = 0 \cdot u

Simplify

uu - \frac{1}{u} u - 23 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{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

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 1 - 23 u = 0

u^{2} - 1 - 23 u = 0

u^{2} - 1 - 23 u = 0

Solve

u^{2} - 1 - 23 u = 0 : u = 3 + 2, u = 3 - 2

u^{2} - 1 - 23 u = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} - 23 u - 1 = 0

Solve with the quadratic formula

u^{2} - 23 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 23, c = - 1

u_{1, 2} = \frac{- ( - 2 3 ) \pm ( - 2 3 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 2 3 ) \pm ( - 2 3 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 23)^{2} - 4 \cdot 1 \cdot (- 1) = 4

(- 23)^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= (- 23)^{2} + 4 \cdot 1 \cdot 1

(- 23)^{2} = 2^{2} \cdot 3

(- 23)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 23)^{2} = (23)^{2}

= (23)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (3)^{2}

(3)^{2} : 3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 2^{2} \cdot 3

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 4

= 2^{2} \cdot 3 + 4

2^{2} \cdot 3 = 12

2^{2} \cdot 3

2^{2} = 4

= 4 \cdot 3

Multiply the numbers:

4 \cdot 3 = 12

= 12

= 12 + 4

Add the numbers:

12 + 4 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 2 3 ) \pm 4}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 2 3 ) + 4}{2 \cdot 1}, u_{2} = \frac{- ( - 2 3 ) - 4}{2 \cdot 1}

u = \frac{- ( - 2 3 ) + 4}{2 \cdot 1} : 3 + 2

\frac{- ( - 2 3 ) + 4}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 3 + 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 3 + 4}{2}

Factor

23 + 4 : 2 (3 + 2)

23 + 4

Rewrite as

= 23 + 2 \cdot 2

Factor out common term

2

= 2 (3 + 2)

= \frac{2 ( 3 + 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 + 2

u = \frac{- ( - 2 3 ) - 4}{2 \cdot 1} : 3 - 2

\frac{- ( - 2 3 ) - 4}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 3 - 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 3 - 4}{2}

Factor

23 - 4 : 2 (3 - 2)

23 - 4

Rewrite as

= 23 - 2 \cdot 2

Factor out common term

2

= 2 (3 - 2)

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

Divide the numbers:

\frac{2}{2} = 1

= 3 - 2

The solutions to the quadratic equation are:

u = 3 + 2, u = 3 - 2

u = 3 + 2, u = 3 - 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - \frac{1}{u} - 23

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3 + 2, u = 3 - 2

Substitute back

u = cot (x)

cot (x) = 3 + 2, cot (x) = 3 - 2

cot (x) = 3 + 2, cot (x) = 3 - 2

cot (x) = 3 + 2 : x = arccot (3 + 2) + πn

cot (x) = 3 + 2

Apply trig inverse properties

cot (x) = 3 + 2

General solutions for

cot (x) = 3 + 2

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

x = arccot (3 + 2) + πn

x = arccot (3 + 2) + πn

cot (x) = 3 - 2 : x = arccot (3 - 2) + πn

cot (x) = 3 - 2

Apply trig inverse properties

cot (x) = 3 - 2

General solutions for

cot (x) = 3 - 2

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

x = arccot (3 - 2) + πn

x = arccot (3 - 2) + πn

Combine all the solutions

x = arccot (3 + 2) + πn, x = arccot (3 - 2) + πn

Show solutions in decimal form

x = 0.26179 \dots + πn, x = 1.83259 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cot(x)-tan(x)=2sqrt(3) ?
The general solution for cot(x)-tan(x)=2sqrt(3) is x=0.26179…+pin,x=1.83259…+pin