What is the general solution for solvefor y,x=-3|sin(y)| ?

The general solution for solvefor y,x=-3|sin(y)| is No Solution for y\in\mathbb{R}

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Popular >

solvefor y,x=-3|sin(y)|

Solution

solve for

y, x = - 3 ∣ sin (y) ∣

Solution

No Solution for y \in R

Solution steps

x = - 3 ∣ sin (y) ∣

Switch sides

- 3 ∣ sin (y) ∣ = x

Solve by substitution

- 3 ∣ sin (y) ∣ = x

Let:

sin (y) = u

- 3 ∣ u ∣ = x

- 3 ∣ u ∣ = x :

No Solution for

u \in R

- 3 ∣ u ∣ = x

Divide both sides by

- 3

- 3 ∣ u ∣ = x

Divide both sides by

- 3

\frac{- 3 ∣ u ∣}{- 3} = \frac{x}{- 3}

Simplify

∣ u ∣ = - \frac{x}{3}

∣ u ∣ = - \frac{x}{3}

Find positive and negative intervals

Find intervals for

∣ u ∣

u \geq 0

u \geq 0, ∣ u ∣ = u

Rewrite

∣ u ∣

for

u \geq 0 : ∣ u ∣ = u

Apply absolute rule: If

u \geq 0

then

∣ u ∣ = u

∣ u ∣ = u

u < 0

u < 0, ∣ u ∣ = - u

Rewrite

∣ u ∣

for

u < 0 : ∣ u ∣ = - u

Apply absolute rule: If

u < 0

then

∣ u ∣ = - u

∣ u ∣ = - u

Identify the intervals:

u < 0, u \geq 0

∣ u ∣ u < 0 - u \geq 0 +

u < 0, u \geq 0

u < 0, u \geq 0

Solve the inequality for each interval

u < 0, u \geq 0

For

u < 0 :

No Solution

For

u < 0

rewrite

∣ u ∣ = - \frac{x}{3}

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

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

\frac{- u}{- 1} = \frac{- \frac{x}{3}}{- 1}

Simplify

\frac{- u}{- 1} = \frac{- \frac{x}{3}}{- 1}

Simplify

\frac{- u}{- 1} : u

\frac{- u}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{u}{1}

Apply rule

\frac{a}{1} = a

= u

Simplify

\frac{- \frac{x}{3}}{- 1} : \frac{x}{3}

\frac{- \frac{x}{3}}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{\frac{x}{3}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{x}{3}

u = \frac{x}{3}

u = \frac{x}{3}

u = \frac{x}{3}

Combine the intervals

No Solution and u < 0

Merge Overlapping Intervals

No Solution and u < 0

The intersection of two intervals is the set of numbers which are in both intervals

No Solution

and

u < 0

No Solution

No Solution

For

u \geq 0 :

No Solution

For

u \geq 0

rewrite

∣ u ∣ = - \frac{x}{3}

u = - \frac{x}{3}

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

u = - \frac{x}{3}

Could not simplify further

u = - \frac{x}{3}

Combine the intervals

No Solution and u \geq 0

Merge Overlapping Intervals

No Solution and u \geq 0

The intersection of two intervals is the set of numbers which are in both intervals

No Solution

and

u \geq 0

No Solution

No Solution

Combine Solutions:

No Solution

No Solution for u \in R

Substitute back

u = sin (y)

No Solution for sin (y) \in R

No Solution for sin (y) \in R

sin (y) =

No Solution

: y = arcsin (No Solution) + 2 πn, y = π + arcsin (- No Solution) + 2 πn

sin (y) = No Solution

Apply trig inverse properties

sin (y) = No Solution

General solutions for

sin (y) =

No Solution

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

y = arcsin (No Solution) + 2 πn, y = π + arcsin (- No Solution) + 2 πn

y = arcsin (No Solution) + 2 πn, y = π + arcsin (- No Solution) + 2 πn

Combine all the solutions

y = arcsin (No Solution) + 2 πn, y = π + arcsin (- No Solution) + 2 πn

Since the equation is undefined for:

arcsin (No Solution) + 2 πn, π + arcsin (- No Solution) + 2 πn

No Solution for y \in R

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor y,x=-3|sin(y)| ?
The general solution for solvefor y,x=-3|sin(y)| is No Solution for y\in\mathbb{R}