What is the general solution for solvefor y,x=sqrt(sin(y)) ?

The general solution for solvefor y,x=sqrt(sin(y)) is y=arcsin(x^2)+2pin,y=pi+arcsin(-x^2)+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

solvefor y,x=sqrt(sin(y))

Solution

solve for

y, x = sin (y)

Solution

y = arcsin (x^{2}) + 2 πn, y = π + arcsin (- x^{2}) + 2 πn

Solution steps

x = sin (y)

Switch sides

sin (y) = x

Square both sides:

sin (y) = x^{2}

sin (y) = x

(sin (y))^{2} = x^{2}

Expand

(sin (y))^{2} : sin (y)

(sin (y))^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (sin^{\frac{1}{2}} (y))^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= sin^{\frac{1}{2} \cdot 2} (y)

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= sin (y)

sin (y) = x^{2}

sin (y) = x^{2}

Verify Solutions:

sin (y) = x^{2} {x \geq 0}

Check the solutions by plugging them into

sin (y) = x

Remove the ones that don't agree with the equation.

Plug

sin (y) = x^{2} : x^{2} = x \Rightarrow x \geq 0

x^{2} = x

Square both sides:

x^{2} = x^{2}

x^{2} = x

(x^{2})^{2} = x^{2}

Expand

(x^{2})^{2} : x^{2}

(x^{2})^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((x^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (x^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= x^{2}

x^{2} = x^{2}

x^{2} = x^{2}

Both sides are equal

True for all x

Verify Solutions:

x < 0

False

, x = 0

True

, x > 0

True

x^{2} = x

Combine domain interval with solution interval:

True for all x

Find the function intervals:

x < 0, x = 0, x > 0

x^{2} = x

Find the even roots arguments zeroes:

Solve

x^{2} = 0 : x = 0

x^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

x = 0

x = 0

The intervals are defined around the zeroes:

x < 0, x = 0, x > 0

Combine intervals with domain

x < 0, x = 0, x > 0

Check the solutions by plugging them into

x^{2} = x

Remove the ones that don't agree with the equation.

Plug

x < 0 : x^{2} \neq = x \Rightarrow

False

The solution is

x \geq 0

The solution is

sin (y) = x^{2} {x \geq 0}

Apply trig inverse properties

sin (y) = x^{2}

General solutions for

sin (y) = x^{2}

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

y = arcsin (x^{2}) + 2 πn, y = π + arcsin (- x^{2}) + 2 πn

y = arcsin (x^{2}) + 2 πn, y = π + arcsin (- x^{2}) + 2 πn

Popular Examples

sin(x)+sin(x)cos(x)=0

sin (x) + sin (x) cos (x) = 0

tan(x)= 5/5

tan (x) = \frac{5}{5}

sin(θ)cos(θ)=0

sin (θ) cos (θ) = 0

csc(θ)= 1/2

csc (θ) = \frac{1}{2}

sin(θ)= 3/2

sin (θ) = \frac{3}{2}

Frequently Asked Questions (FAQ)

What is the general solution for solvefor y,x=sqrt(sin(y)) ?
The general solution for solvefor y,x=sqrt(sin(y)) is y=arcsin(x^2)+2pin,y=pi+arcsin(-x^2)+2pin