What is the general solution for 5cot(x)+3tan(x)=8 ?

The general solution for 5cot(x)+3tan(x)=8 is x= pi/4+pin,x=1.03037…+pin

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5cot(x)+3tan(x)=8

Solution

5 cot (x) + 3 tan (x) = 8

Solution

x = \frac{π}{4} + πn, x = 1.03037 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 59.03624 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 cot (x) + 3 tan (x) = 8

Subtract

8

from both sides

5 cot (x) + 3 tan (x) - 8 = 0

Rewrite using trig identities

- 8 + 3 tan (x) + 5 cot (x)

Use the basic trigonometric identity:

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

= - 8 + 3 \cdot \frac{1}{cot ( x )} + 5 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 )}

= - 8 + \frac{3}{cot ( x )} + 5 cot (x)

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

Solve by substitution

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

Let:

cot (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

5 uu : 5 u^{2}

5 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 5 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 5 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

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

5 u^{2} - 8 u + 3 = 0

Solve with the quadratic formula

5 u^{2} - 8 u + 3 = 0

Quadratic Equation Formula:

For

a = 5, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

(- 8)^{2} - 4 \cdot 5 \cdot 3 = 2

(- 8)^{2} - 4 \cdot 5 \cdot 3

Apply exponent rule:

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

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

Subtract the numbers:

64 - 60 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 8 ) \pm 2}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 2}{2 \cdot 5}, u_{2} = \frac{- ( - 8 ) - 2}{2 \cdot 5}

u = \frac{- ( - 8 ) + 2}{2 \cdot 5} : 1

\frac{- ( - 8 ) + 2}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{8 + 2}{2 \cdot 5}

Add the numbers:

8 + 2 = 10

= \frac{10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 8 ) - 2}{2 \cdot 5} : \frac{3}{5}

\frac{- ( - 8 ) - 2}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{8 - 2}{2 \cdot 5}

Subtract the numbers:

8 - 2 = 6

= \frac{6}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{6}{10}

Cancel the common factor:

2

= \frac{3}{5}

The solutions to the quadratic equation are:

u = 1, u = \frac{3}{5}

u = 1, u = \frac{3}{5}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = \frac{3}{5}

Substitute back

u = cot (x)

cot (x) = 1, cot (x) = \frac{3}{5}

cot (x) = 1, cot (x) = \frac{3}{5}

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

cot (x) = \frac{3}{5} : x = arccot (\frac{3}{5}) + πn

cot (x) = \frac{3}{5}

Apply trig inverse properties

cot (x) = \frac{3}{5}

General solutions for

cot (x) = \frac{3}{5}

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

x = arccot (\frac{3}{5}) + πn

x = arccot (\frac{3}{5}) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arccot (\frac{3}{5}) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = 1.03037 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5cot(x)+3tan(x)=8 ?
The general solution for 5cot(x)+3tan(x)=8 is x= pi/4+pin,x=1.03037…+pin