What is the general solution for 4tan(x)+cot(x)=5 ?

The general solution for 4tan(x)+cot(x)=5 is x=0.24497…+pin,x= pi/4+pin

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Popular Trigonometry >

4tan(x)+cot(x)=5

Solution

4 tan (x) + cot (x) = 5

Solution

x = 0.24497 \dots + πn, x = \frac{π}{4} + πn

Degrees

x = 14.03624 \dots^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

4 tan (x) + cot (x) = 5

Subtract

5

from both sides

4 tan (x) + cot (x) - 5 = 0

Rewrite using trig identities

- 5 + cot (x) + 4 tan (x)

Use the basic trigonometric identity:

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

= - 5 + cot (x) + 4 \cdot \frac{1}{cot ( x )}

4 \cdot \frac{1}{cot ( x )} = \frac{4}{cot ( x )}

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

Multiply fractions:

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

= \frac{1 \cdot 4}{cot ( x )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{cot ( x )}

= - 5 + cot (x) + \frac{4}{cot ( x )}

- 5 + cot (x) + \frac{4}{cot ( x )} = 0

Solve by substitution

- 5 + cot (x) + \frac{4}{cot ( x )} = 0

Let:

cot (x) = u

- 5 + u + \frac{4}{u} = 0

- 5 + u + \frac{4}{u} = 0 : u = 4, u = 1

- 5 + u + \frac{4}{u} = 0

Multiply both sides by

u

- 5 + u + \frac{4}{u} = 0

Multiply both sides by

u

- 5 u + uu + \frac{4}{u} u = 0 \cdot u

Simplify

- 5 u + uu + \frac{4}{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{4}{u} u : 4

\frac{4}{u} u

Multiply fractions:

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

= \frac{4 u}{u}

Cancel the common factor:

u

= 4

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

- 5 u + u^{2} + 4 = 0 : u = 4, u = 1

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = - 5, c = 4

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

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

(- 5)^{2} - 4 \cdot 1 \cdot 4 = 3

(- 5)^{2} - 4 \cdot 1 \cdot 4

Apply exponent rule:

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

n

is even

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

= 5^{2} - 4 \cdot 1 \cdot 4

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

Subtract the numbers:

25 - 16 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

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

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 1}

u = \frac{- ( - 5 ) + 3}{2 \cdot 1} : 4

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

Apply rule

- (- a) = a

= \frac{5 + 3}{2 \cdot 1}

Add the numbers:

5 + 3 = 8

= \frac{8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

u = \frac{- ( - 5 ) - 3}{2 \cdot 1} : 1

\frac{- ( - 5 ) - 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 - 3}{2 \cdot 1}

Subtract the numbers:

5 - 3 = 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = 4, u = 1

u = 4, u = 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 5 + u + \frac{4}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 4, u = 1

Substitute back

u = cot (x)

cot (x) = 4, cot (x) = 1

cot (x) = 4, cot (x) = 1

cot (x) = 4 : x = arccot (4) + πn

cot (x) = 4

Apply trig inverse properties

cot (x) = 4

General solutions for

cot (x) = 4

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

x = arccot (4) + πn

x = arccot (4) + πn

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

General solutions for

cot (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = arccot (4) + πn, x = \frac{π}{4} + πn

Show solutions in decimal form

x = 0.24497 \dots + πn, x = \frac{π}{4} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan(x)+cot(x)=5 ?
The general solution for 4tan(x)+cot(x)=5 is x=0.24497…+pin,x= pi/4+pin