What is the general solution for 12tan(θ)+5=5tan(θ)+5 ?

The general solution for 12tan(θ)+5=5tan(θ)+5 is θ=pin

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12tan(θ)+5=5tan(θ)+5

12 tan (θ) + 5 = 5 tan (θ) + 5

θ = πn

Degrees

θ = 0^{\circ} + 18 0^{\circ} n

Solution steps

12 tan (θ) + 5 = 5 tan (θ) + 5

Solve by substitution

12 tan (θ) + 5 = 5 tan (θ) + 5

Let:

tan (θ) = u

12 u + 5 = 5 u + 5

12 u + 5 = 5 u + 5 : u = 0

12 u + 5 = 5 u + 5

Subtract

5

from both sides

12 u + 5 - 5 = 5 u + 5 - 5

Simplify

12 u = 5 u

Move

5 u

to the left side

12 u = 5 u

Subtract

5 u

from both sides

12 u - 5 u = 5 u - 5 u

Simplify

7 u = 0

7 u = 0

Divide both sides by

7

7 u = 0

Divide both sides by

7

\frac{7 u}{7} = \frac{0}{7}

Simplify

u = 0

u = 0

Substitute back

u = tan (θ)

tan (θ) = 0

tan (θ) = 0

tan (θ) = 0 : θ = πn

tan (θ) = 0

General solutions for

tan (θ) = 0

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 = 0 + πn

θ = 0 + πn

Solve

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

Combine all the solutions

θ = πn

1 - sin (x) - 2 sin^{2} (x) = 0

sin^{2} (x) + cos^{2} (x) - 1 + cos (x) - sin (x) = 0

sin (4 x) - cos (2 x) = 0

\frac{6}{π} = 3 cos (2 x)

sin (2 x) = - 2 cos (x)

What is the general solution for 12tan(θ)+5=5tan(θ)+5 ?
The general solution for 12tan(θ)+5=5tan(θ)+5 is θ=pin