What is the general solution for 5sin(θ)-4=2sin(θ)-6 ?

The general solution for 5sin(θ)-4=2sin(θ)-6 is θ=-0.72972…+2pin,θ=pi+0.72972…+2pin

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5sin(θ)-4=2sin(θ)-6

5 sin (θ) - 4 = 2 sin (θ) - 6

θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

Degrees

θ = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 221.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin (θ) - 4 = 2 sin (θ) - 6

Solve by substitution

5 sin (θ) - 4 = 2 sin (θ) - 6

Let:

sin (θ) = u

5 u - 4 = 2 u - 6

5 u - 4 = 2 u - 6 : u = - \frac{2}{3}

5 u - 4 = 2 u - 6

Move

4

to the right side

5 u - 4 = 2 u - 6

Add

4

to both sides

5 u - 4 + 4 = 2 u - 6 + 4

Simplify

5 u = 2 u - 2

5 u = 2 u - 2

Move

2 u

to the left side

5 u = 2 u - 2

Subtract

2 u

from both sides

5 u - 2 u = 2 u - 2 - 2 u

Simplify

3 u = - 2

3 u = - 2

Divide both sides by

3

3 u = - 2

Divide both sides by

3

\frac{3 u}{3} = \frac{- 2}{3}

Simplify

u = - \frac{2}{3}

u = - \frac{2}{3}

Substitute back

u = sin (θ)

sin (θ) = - \frac{2}{3}

sin (θ) = - \frac{2}{3}

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

sin (θ) = - \frac{2}{3}

Apply trig inverse properties

sin (θ) = - \frac{2}{3}

General solutions for

sin (θ) = - \frac{2}{3}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

cos (2 θ) - cos (4 θ) = 0

tan (x) = 1.4

sin (θ) = 0.71

2 tan^{2} (x) - 3 tan (x) + 1 = 0

tan (x) = \frac{8}{6}

What is the general solution for 5sin(θ)-4=2sin(θ)-6 ?
The general solution for 5sin(θ)-4=2sin(θ)-6 is θ=-0.72972…+2pin,θ=pi+0.72972…+2pin