What is the general solution for 5tan(x)+3=4(1+tan(x)) ?

The general solution for 5tan(x)+3=4(1+tan(x)) is x= pi/4+pin

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5tan(x)+3=4(1+tan(x))

5 tan (x) + 3 = 4 (1 + tan (x))

x = \frac{π}{4} + πn

Degrees

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

Solution steps

5 tan (x) + 3 = 4 (1 + tan (x))

Solve by substitution

5 tan (x) + 3 = 4 (1 + tan (x))

Let:

tan (x) = u

5 u + 3 = 4 (1 + u)

5 u + 3 = 4 (1 + u) : u = 1

5 u + 3 = 4 (1 + u)

Expand

4 (1 + u) : 4 + 4 u

4 (1 + u)

Apply the distributive law:

a (b + c) = ab + a c

a = 4, b = 1, c = u

= 4 \cdot 1 + 4 u

Multiply the numbers:

4 \cdot 1 = 4

= 4 + 4 u

5 u + 3 = 4 + 4 u

Move

3

to the right side

5 u + 3 = 4 + 4 u

Subtract

3

from both sides

5 u + 3 - 3 = 4 + 4 u - 3

Simplify

5 u = 4 u + 1

5 u = 4 u + 1

Move

4 u

to the left side

5 u = 4 u + 1

Subtract

4 u

from both sides

5 u - 4 u = 4 u + 1 - 4 u

Simplify

u = 1

u = 1

Substitute back

u = tan (x)

tan (x) = 1

tan (x) = 1

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn

cos (x) + 2 = 3

9 sin (b) + 3 = 5 sin (b) + 1

4 tan (x) + 2 sin (x) cos (x) = 0

cos (x) = 2 cos^{2} (x) - 1

3 sin (x) + 6 cos (x) = 4

What is the general solution for 5tan(x)+3=4(1+tan(x)) ?
The general solution for 5tan(x)+3=4(1+tan(x)) is x= pi/4+pin