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

The general solution for sin(θ)+3=5sin(θ) is θ=0.84806…+2pin,θ=pi-0.84806…+2pin

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sin(θ)+3=5sin(θ)

sin (θ) + 3 = 5 sin (θ)

θ = 0.84806 \dots + 2 πn, θ = π - 0.84806 \dots + 2 πn

Degrees

θ = 48.59037 \dots^{\circ} + 36 0^{\circ} n, θ = 131.40962 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) + 3 = 5 sin (θ)

Solve by substitution

sin (θ) + 3 = 5 sin (θ)

Let:

sin (θ) = u

u + 3 = 5 u

u + 3 = 5 u : u = \frac{3}{4}

u + 3 = 5 u

Move

3

to the right side

u + 3 = 5 u

Subtract

3

from both sides

u + 3 - 3 = 5 u - 3

Simplify

u = 5 u - 3

u = 5 u - 3

Move

5 u

to the left side

u = 5 u - 3

Subtract

5 u

from both sides

u - 5 u = 5 u - 3 - 5 u

Simplify

- 4 u = - 3

- 4 u = - 3

Divide both sides by

- 4

- 4 u = - 3

Divide both sides by

- 4

\frac{- 4 u}{- 4} = \frac{- 3}{- 4}

Simplify

u = \frac{3}{4}

u = \frac{3}{4}

Substitute back

u = sin (θ)

sin (θ) = \frac{3}{4}

sin (θ) = \frac{3}{4}

sin (θ) = \frac{3}{4} : θ = arcsin (\frac{3}{4}) + 2 πn, θ = π - arcsin (\frac{3}{4}) + 2 πn

sin (θ) = \frac{3}{4}

Apply trig inverse properties

sin (θ) = \frac{3}{4}

General solutions for

sin (θ) = \frac{3}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

θ = arcsin (\frac{3}{4}) + 2 πn, θ = π - arcsin (\frac{3}{4}) + 2 πn

θ = arcsin (\frac{3}{4}) + 2 πn, θ = π - arcsin (\frac{3}{4}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{3}{4}) + 2 πn, θ = π - arcsin (\frac{3}{4}) + 2 πn

Show solutions in decimal form

θ = 0.84806 \dots + 2 πn, θ = π - 0.84806 \dots + 2 πn

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

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

0.87 = 5 cos (3 x)

cos (\frac{2 π x}{3}) = \frac{1}{2}

- \frac{1}{2} = cos (2 x)

What is the general solution for sin(θ)+3=5sin(θ) ?
The general solution for sin(θ)+3=5sin(θ) is θ=0.84806…+2pin,θ=pi-0.84806…+2pin