What is the general solution for (2-3sin(θ))/2 =-3sin(θ) ?

The general solution for (2-3sin(θ))/2 =-3sin(θ) is θ=-0.72972…+2pin,θ=pi+0.72972…+2pin

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Popular >

(2-3sin(θ))/2 =-3sin(θ)

Solution

\frac{2 - 3 sin ( θ )}{2} = - 3 sin (θ)

Solution

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

\frac{2 - 3 sin ( θ )}{2} = - 3 sin (θ)

Solve by substitution

\frac{2 - 3 sin ( θ )}{2} = - 3 sin (θ)

Let:

sin (θ) = u

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

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 - 3 u}{2} \cdot 2 : 2 - 3 u

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 2 - 3 u ) \cdot 2}{2}

Cancel the common factor:

2

= 2 - 3 u

Simplify

- 3 u \cdot 2 : - 6 u

- 3 u \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= - 6 u

2 - 3 u = - 6 u

2 - 3 u = - 6 u

2 - 3 u = - 6 u

Move

2

to the right side

2 - 3 u = - 6 u

Subtract

2

from both sides

2 - 3 u - 2 = - 6 u - 2

Simplify

- 3 u = - 6 u - 2

- 3 u = - 6 u - 2

Move

6 u

to the left side

- 3 u = - 6 u - 2

Add

6 u

to both sides

- 3 u + 6 u = - 6 u - 2 + 6 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

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Frequently Asked Questions (FAQ)

What is the general solution for (2-3sin(θ))/2 =-3sin(θ) ?
The general solution for (2-3sin(θ))/2 =-3sin(θ) is θ=-0.72972…+2pin,θ=pi+0.72972…+2pin