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

The general solution for 5sqrt(3)tan(x)+3=8sqrt(3)tan(x) is x= pi/6+pin

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Popular >

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

Solution

53 tan (x) + 3 = 83 tan (x)

Solution

x = \frac{π}{6} + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

53 tan (x) + 3 = 83 tan (x)

Solve by substitution

53 tan (x) + 3 = 83 tan (x)

Let:

tan (x) = u

53 u + 3 = 83 u

53 u + 3 = 83 u : u = \frac{3}{3}

53 u + 3 = 83 u

Move

3

to the right side

53 u + 3 = 83 u

Subtract

3

from both sides

53 u + 3 - 3 = 83 u - 3

Simplify

53 u = 83 u - 3

53 u = 83 u - 3

Move

83 u

to the left side

53 u = 83 u - 3

Subtract

83 u

from both sides

53 u - 83 u = 83 u - 3 - 83 u

Simplify

53 u - 83 u = 83 u - 3 - 83 u

Simplify

53 u - 83 u : - 33 u

53 u - 83 u

Add similar elements:

53 u - 83 u = - 33 u

= - 33 u

Simplify

83 u - 3 - 83 u : - 3

83 u - 3 - 83 u

Add similar elements:

83 u - 83 u = 0

= - 3

- 33 u = - 3

- 33 u = - 3

- 33 u = - 3

Divide both sides by

- 33

- 33 u = - 3

Divide both sides by

- 33

\frac{- 3 3 u}{- 3 3} = \frac{- 3}{- 3 3}

Simplify

\frac{- 3 3 u}{- 3 3} = \frac{- 3}{- 3 3}

Simplify

\frac{- 3 3 u}{- 3 3} : u

\frac{- 3 3 u}{- 3 3}

Apply the fraction rule:

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

= \frac{3 3 u}{3 3}

Divide the numbers:

\frac{3}{3} = 1

= \frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 3}{- 3 3} : \frac{3}{3}

\frac{- 3}{- 3 3}

Apply the fraction rule:

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

= \frac{3}{3 3}

Divide the numbers:

\frac{3}{3} = 1

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

Substitute back

u = tan (x)

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3} : x = \frac{π}{6} + πn

tan (x) = \frac{3}{3}

General solutions for

tan (x) = \frac{3}{3}

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = \frac{π}{6} + πn

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\frac{π}{12} = arcsin (\frac{x}{2})

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Frequently Asked Questions (FAQ)

What is the general solution for 5sqrt(3)tan(x)+3=8sqrt(3)tan(x) ?
The general solution for 5sqrt(3)tan(x)+3=8sqrt(3)tan(x) is x= pi/6+pin