What is the general solution for 2tan(θ)+4=tan(θ)+5 ?

The general solution for 2tan(θ)+4=tan(θ)+5 is θ= pi/4+pin

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2tan(θ)+4=tan(θ)+5

2 tan (θ) + 4 = tan (θ) + 5

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

2 tan (θ) + 4 = tan (θ) + 5

Solve by substitution

2 tan (θ) + 4 = tan (θ) + 5

Let:

tan (θ) = u

2 u + 4 = u + 5

2 u + 4 = u + 5 : u = 1

2 u + 4 = u + 5

Move

4

to the right side

2 u + 4 = u + 5

Subtract

4

from both sides

2 u + 4 - 4 = u + 5 - 4

Simplify

2 u = u + 1

2 u = u + 1

Move

u

to the left side

2 u = u + 1

Subtract

u

from both sides

2 u - u = u + 1 - u

Simplify

u = 1

u = 1

Substitute back

u = tan (θ)

tan (θ) = 1

tan (θ) = 1

tan (θ) = 1 : θ = \frac{π}{4} + πn

tan (θ) = 1

General solutions for

tan (θ) = 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}

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

Combine all the solutions

θ = \frac{π}{4} + πn

What is the general solution for 2tan(θ)+4=tan(θ)+5 ?
The general solution for 2tan(θ)+4=tan(θ)+5 is θ= pi/4+pin