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

The general solution for 3tan(x)-2cot(x)=5 is x=2.81984…+pin,x=1.10714…+pin

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

Solution

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

Solution

x = 2.81984 \dots + πn, x = 1.10714 \dots + πn

Degrees

x = 161.56505 \dots^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

5

from both sides

3 tan (x) - 2 cot (x) - 5 = 0

Rewrite using trig identities

- 5 - 2 cot (x) + 3 tan (x)

Use the basic trigonometric identity:

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

= - 5 - 2 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 )}

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

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

Solve by substitution

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

Let:

cot (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

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

- 2 uu : - 2 u^{2}

- 2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 2 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 2 \cdot 3 = 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

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

Separate the solutions

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

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

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

Remove parentheses:

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

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

Add the numbers:

5 + 7 = 12

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{12}{- 4}

Apply the fraction rule:

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

= - \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= - 3

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

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

Remove parentheses:

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

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

Subtract the numbers:

5 - 7 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

Apply the fraction rule:

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

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = cot (x)

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

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

cot (x) = - 3 : x = arccot (- 3) + πn

cot (x) = - 3

Apply trig inverse properties

cot (x) = - 3

General solutions for

cot (x) = - 3

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

x = arccot (- 3) + πn

x = arccot (- 3) + πn

cot (x) = \frac{1}{2} : x = arccot (\frac{1}{2}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (\frac{1}{2}) + πn

x = arccot (\frac{1}{2}) + πn

Combine all the solutions

x = arccot (- 3) + πn, x = arccot (\frac{1}{2}) + πn

Show solutions in decimal form

x = 2.81984 \dots + πn, x = 1.10714 \dots + πn

Popular Examples

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tan (θ) = \frac{\frac{3}{1} - \frac{2}{5}}{1 + \frac{3}{1} \cdot \frac{2}{5}}

tan(t)=14

tan (t) = 14

cos(A)= 5/8

cos (A) = \frac{5}{8}

sin(9x+40)=cos(-5x+42)

sin (9 x + 40) = cos (- 5 x + 42)

sin^2(a)=-((5sqrt(11)))/((18))

sin^{2} (a) = - \frac{( 5 11 )}{( 18 )}

Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x)-2cot(x)=5 ?
The general solution for 3tan(x)-2cot(x)=5 is x=2.81984…+pin,x=1.10714…+pin