What is the general solution for solvefor y,x=5*sin(sqrt(y)) ?

The general solution for solvefor y,x=5*sin(sqrt(y)) is

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solvefor y,x=5*sin(sqrt(y))

Solution

solve for

y, x = 5 \cdot sin (y)

Solution

y = arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}, y = π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

Solution steps

x = 5 sin (y)

Switch sides

5 sin (y) = x

Divide both sides by

5

5 sin (y) = x

Divide both sides by

5

\frac{5 sin ( y )}{5} = \frac{x}{5}

Simplify

sin (y) = \frac{x}{5}

sin (y) = \frac{x}{5}

Apply trig inverse properties

sin (y) = \frac{x}{5}

General solutions for

sin (y) = \frac{x}{5}

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

y = arcsin (\frac{x}{5}) + 2 πn, y = π + arcsin (- \frac{x}{5}) + 2 πn

y = arcsin (\frac{x}{5}) + 2 πn, y = π + arcsin (- \frac{x}{5}) + 2 πn

Solve

y = arcsin (\frac{x}{5}) + 2 πn : y = arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

y = arcsin (\frac{x}{5}) + 2 πn

Square both sides:

y = arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

y = arcsin (\frac{x}{5}) + 2 πn

(y)^{2} = (arcsin (\frac{x}{5}) + 2 πn)^{2}

Expand

(y)^{2} : y

(y)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= y^{\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

= y

Expand

(arcsin (\frac{x}{5}) + 2 πn)^{2} : arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

(arcsin (\frac{x}{5}) + 2 πn)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = arcsin (\frac{x}{5}), b = 2 πn

= arcsin^{2} (\frac{x}{5}) + 2 arcsin (\frac{x}{5}) \cdot 2 πn + (2 πn)^{2}

Simplify

arcsin^{2} (\frac{x}{5}) + 2 arcsin (\frac{x}{5}) \cdot 2 πn + (2 πn)^{2} : arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

arcsin^{2} (\frac{x}{5}) + 2 arcsin (\frac{x}{5}) \cdot 2 πn + (2 πn)^{2}

2 arcsin (\frac{x}{5}) \cdot 2 πn = 4 πn arcsin (\frac{x}{5})

2 arcsin (\frac{x}{5}) \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn arcsin (\frac{x}{5})

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

= arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

y = arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

y = arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}

Solve

y = π + arcsin (- \frac{x}{5}) + 2 πn : y = π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

y = π + arcsin (- \frac{x}{5}) + 2 πn

Square both sides:

y = π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

y = π + arcsin (- \frac{x}{5}) + 2 πn

(y)^{2} = (π + arcsin (- \frac{x}{5}) + 2 πn)^{2}

Expand

(y)^{2} : y

(y)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= y^{\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

= y

Expand

(π + arcsin (- \frac{x}{5}) + 2 πn)^{2} : π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

(π + arcsin (- \frac{x}{5}) + 2 πn)^{2}

(π + arcsin (- \frac{x}{5}) + 2 πn)^{2} = (π + arcsin (- \frac{x}{5}) + 2 πn) (π + arcsin (- \frac{x}{5}) + 2 πn)

= (π + 2 πn + arcsin (- \frac{x}{5})) (π + 2 πn + arcsin (- \frac{x}{5}))

Expand

(π + arcsin (- \frac{x}{5}) + 2 πn) (π + arcsin (- \frac{x}{5}) + 2 πn) : π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

(π + arcsin (- \frac{x}{5}) + 2 πn) (π + arcsin (- \frac{x}{5}) + 2 πn)

Distribute parentheses

= ππ + π arcsin (- \frac{x}{5}) + π 2 πn + arcsin (- \frac{x}{5}) π + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) \cdot 2 πn + 2 πnπ + 2 πn arcsin (- \frac{x}{5}) + 2 πn \cdot 2 πn

= ππ + π arcsin (- \frac{x}{5}) + 2 ππn + π arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5}) + 2 πn arcsin (- \frac{x}{5}) + 2 ππn + 2 πn arcsin (- \frac{x}{5}) + 2 \cdot 2 ππnn

Simplify

ππ + π arcsin (- \frac{x}{5}) + 2 ππn + π arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5}) + 2 πn arcsin (- \frac{x}{5}) + 2 ππn + 2 πn arcsin (- \frac{x}{5}) + 2 \cdot 2 ππnn : π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

ππ + π arcsin (- \frac{x}{5}) + 2 ππn + π arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5}) + 2 πn arcsin (- \frac{x}{5}) + 2 ππn + 2 πn arcsin (- \frac{x}{5}) + 2 \cdot 2 ππnn

Group like terms

= ππ + 2 ππn + 2 ππn + 2 \cdot 2 ππnn + π arcsin (- \frac{x}{5}) + π arcsin (- \frac{x}{5}) + 2 πn arcsin (- \frac{x}{5}) + 2 πn arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5})

Add similar elements:

2 πn arcsin (- \frac{x}{5}) + 2 πn arcsin (- \frac{x}{5}) = 4 πn arcsin (- \frac{x}{5})

= ππ + 2 ππn + 2 ππn + 2 \cdot 2 ππnn + π arcsin (- \frac{x}{5}) + π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5})

Add similar elements:

π arcsin (- \frac{x}{5}) + π arcsin (- \frac{x}{5}) = 2 π arcsin (- \frac{x}{5})

= ππ + 2 ππn + 2 ππn + 2 \cdot 2 ππnn + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5})

Add similar elements:

2 ππn + 2 ππn = 4 ππn

= ππ + 4 ππn + 2 \cdot 2 ππnn + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5})

ππ = π^{2}

ππ

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

ππ = π^{1 + 1}

= π^{1 + 1}

Add the numbers:

1 + 1 = 2

= π^{2}

4 ππn = 4 π^{2} n

4 ππn

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

ππ = π^{1 + 1}

= 4 π^{1 + 1} n

Add the numbers:

1 + 1 = 2

= 4 π^{2} n

2 \cdot 2 ππnn = 4 π^{2} n^{2}

2 \cdot 2 ππnn

Multiply the numbers:

2 \cdot 2 = 4

= 4 ππnn

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

nn = n^{1 + 1}

= 4 ππ n^{1 + 1}

Add the numbers:

1 + 1 = 2

= 4 ππ n^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

ππ = π^{1 + 1}

= 4 π^{1 + 1} n^{2}

Add the numbers:

1 + 1 = 2

= 4 π^{2} n^{2}

arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5}) = arcsin^{2} (- \frac{x}{5})

arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5})

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

arcsin (- \frac{x}{5}) arcsin (- \frac{x}{5}) = arcsin^{1 + 1} (- \frac{x}{5})

= arcsin^{1 + 1} (- \frac{x}{5})

Add the numbers:

1 + 1 = 2

= arcsin^{2} (- \frac{x}{5})

= π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

= π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

= π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

y = π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

y = π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

y = arcsin^{2} (\frac{x}{5}) + 4 πn arcsin (\frac{x}{5}) + 4 π^{2} n^{2}, y = π^{2} + 4 π^{2} n + 4 π^{2} n^{2} + 2 π arcsin (- \frac{x}{5}) + 4 πn arcsin (- \frac{x}{5}) + arcsin^{2} (- \frac{x}{5})

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor y,x=5*sin(sqrt(y)) ?
The general solution for solvefor y,x=5*sin(sqrt(y)) is