What is the general solution for -4sin(θ)+5=3sin(θ)+5 ?

The general solution for -4sin(θ)+5=3sin(θ)+5 is θ=2pin,θ=pi+2pin

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-4sin(θ)+5=3sin(θ)+5

- 4 sin (θ) + 5 = 3 sin (θ) + 5

θ = 2 πn, θ = π + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

- 4 sin (θ) + 5 = 3 sin (θ) + 5

Solve by substitution

- 4 sin (θ) + 5 = 3 sin (θ) + 5

Let:

sin (θ) = u

- 4 u + 5 = 3 u + 5

- 4 u + 5 = 3 u + 5 : u = 0

- 4 u + 5 = 3 u + 5

Subtract

5

from both sides

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

Simplify

- 4 u = 3 u

Move

3 u

to the left side

- 4 u = 3 u

Subtract

3 u

from both sides

- 4 u - 3 u = 3 u - 3 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 = sin (θ)

sin (θ) = 0

sin (θ) = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

4 tan^{2} (θ) - 25 = 0

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

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

4 sin (θ) cos^{2} (θ) = 3 sin (θ)

sin (x) = \frac{7}{25}

What is the general solution for -4sin(θ)+5=3sin(θ)+5 ?
The general solution for -4sin(θ)+5=3sin(θ)+5 is θ=2pin,θ=pi+2pin