What is the general solution for 2tan^2(x)-3cot^2(x)=5 ?

The general solution for 2tan^2(x)-3cot^2(x)=5 is x=1.04719…+pin,x=2.09439…+pin

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2tan^2(x)-3cot^2(x)=5

Solution

2 tan^{2} (x) - 3 cot^{2} (x) = 5

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

2 tan^{2} (x) - 3 cot^{2} (x) = 5

Subtract

5

from both sides

2 tan^{2} (x) - 3 cot^{2} (x) - 5 = 0

Rewrite using trig identities

- 5 + 2 tan^{2} (x) - 3 cot^{2} (x)

Use the basic trigonometric identity:

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

= - 5 + 2 (\frac{1}{cot ( x )})^{2} - 3 cot^{2} (x)

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

= - 5 + \frac{2}{cot ^{2} ( x )} - 3 cot^{2} (x)

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

Solve by substitution

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

Let:

cot (x) = u

- 5 + \frac{2}{u ^{2}} - 3 u^{2} = 0

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

- 5 + \frac{2}{u ^{2}} - 3 u^{2} = 0

Multiply both sides by

u^{2}

- 5 + \frac{2}{u ^{2}} - 3 u^{2} = 0

Multiply both sides by

u^{2}

- 5 u^{2} + \frac{2}{u ^{2}} u^{2} - 3 u^{2} u^{2} = 0 \cdot u^{2}

Simplify

- 5 u^{2} + \frac{2}{u ^{2}} u^{2} - 3 u^{2} u^{2} = 0 \cdot u^{2}

Simplify

\frac{2}{u ^{2}} u^{2} : 2

\frac{2}{u ^{2}} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 2

Simplify

- 3 u^{2} u^{2} : - 3 u^{4}

- 3 u^{2} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= - 3 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 3 u^{4}

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- 5 u^{2} + 2 - 3 u^{4} = 0

- 5 u^{2} + 2 - 3 u^{4} = 0

- 5 u^{2} + 2 - 3 u^{4} = 0

Solve

- 5 u^{2} + 2 - 3 u^{4} = 0 : u = 2 i, u = - 2 i, u = \frac{1}{3}, u = - \frac{1}{3}

- 5 u^{2} + 2 - 3 u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 3 u^{4} - 5 u^{2} + 2 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- 3 v^{2} - 5 v + 2 = 0

Solve

- 3 v^{2} - 5 v + 2 = 0 : v = - 2, v = \frac{1}{3}

- 3 v^{2} - 5 v + 2 = 0

Solve with the quadratic formula

- 3 v^{2} - 5 v + 2 = 0

Quadratic Equation Formula:

For

a = - 3, b = - 5, c = 2

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 3 ) \cdot 2}{2 ( - 3 )}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 3 ) \cdot 2}{2 ( - 3 )}

(- 5)^{2} - 4 (- 3) \cdot 2 = 7

(- 5)^{2} - 4 (- 3) \cdot 2

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 3 \cdot 2

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 3 \cdot 2

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

v_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 ( - 3 )}

Separate the solutions

v_{1} = \frac{- ( - 5 ) + 7}{2 ( - 3 )}, v_{2} = \frac{- ( - 5 ) - 7}{2 ( - 3 )}

v = \frac{- ( - 5 ) + 7}{2 ( - 3 )} : - 2

\frac{- ( - 5 ) + 7}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{5 + 7}{- 2 \cdot 3}

Add the numbers:

5 + 7 = 12

= \frac{12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

v = \frac{- ( - 5 ) - 7}{2 ( - 3 )} : \frac{1}{3}

\frac{- ( - 5 ) - 7}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{5 - 7}{- 2 \cdot 3}

Subtract the numbers:

5 - 7 = - 2

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 2}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

The solutions to the quadratic equation are:

v = - 2, v = \frac{1}{3}

v = - 2, v = \frac{1}{3}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = - 2 : u = 2 i, u = - 2 i

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

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

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

Solve

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

The solutions are

u = 2 i, u = - 2 i, u = \frac{1}{3}, u = - \frac{1}{3}

u = 2 i, u = - 2 i, u = \frac{1}{3}, u = - \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 5 + \frac{2}{u ^{2}} - 3 u^{2}

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2 i, u = - 2 i, u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = cot (x)

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

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

cot (x) = 2 i :

No Solution

cot (x) = 2 i

No Solution

cot (x) = - 2 i :

No Solution

cot (x) = - 2 i

No Solution

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

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Frequently Asked Questions (FAQ)

What is the general solution for 2tan^2(x)-3cot^2(x)=5 ?
The general solution for 2tan^2(x)-3cot^2(x)=5 is x=1.04719…+pin,x=2.09439…+pin