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arcsin((900x^2-1)/(900x^2+1))=1.18

Solution

arcsin (\frac{900 x ^{2} - 1}{900 x ^{2} + 1}) = 1.18

Solution

x = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}, x = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

Solution steps

arcsin (\frac{900 x ^{2} - 1}{900 x ^{2} + 1}) = 1.18

Apply trig inverse properties

arcsin (\frac{900 x ^{2} - 1}{900 x ^{2} + 1}) = 1.18

arcsin (x) = a \Rightarrow x = sin (a)

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} = sin (1.18)

sin (1.18) = sin (\frac{59}{50})

sin (1.18)

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} = sin (\frac{59}{50})

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} = sin (\frac{59}{50})

Solve

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} = sin (\frac{59}{50}) : x = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}, x = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} = sin (\frac{59}{50})

Multiply both sides by

900 x^{2} + 1

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} = sin (\frac{59}{50})

Multiply both sides by

900 x^{2} + 1

\frac{900 x ^{2} - 1}{900 x ^{2} + 1} (900 x^{2} + 1) = sin (\frac{59}{50}) (900 x^{2} + 1)

Simplify

900 x^{2} - 1 = sin (\frac{59}{50}) (900 x^{2} + 1)

900 x^{2} - 1 = sin (\frac{59}{50}) (900 x^{2} + 1)

Solve

900 x^{2} - 1 = sin (\frac{59}{50}) (900 x^{2} + 1) : x = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}, x = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

900 x^{2} - 1 = sin (\frac{59}{50}) (900 x^{2} + 1)

Move

1

to the right side

900 x^{2} - 1 = sin (\frac{59}{50}) (900 x^{2} + 1)

Add

1

to both sides

900 x^{2} - 1 + 1 = sin (\frac{59}{50}) (900 x^{2} + 1) + 1

Simplify

900 x^{2} = sin (\frac{59}{50}) (900 x^{2} + 1) + 1

900 x^{2} = sin (\frac{59}{50}) (900 x^{2} + 1) + 1

Move

sin (\frac{59}{50}) (900 x^{2} + 1)

to the left side

900 x^{2} = sin (\frac{59}{50}) (900 x^{2} + 1) + 1

Subtract

sin (\frac{59}{50}) (900 x^{2} + 1)

from both sides

900 x^{2} - sin (\frac{59}{50}) (900 x^{2} + 1) = sin (\frac{59}{50}) (900 x^{2} + 1) + 1 - sin (\frac{59}{50}) (900 x^{2} + 1)

Simplify

900 x^{2} - sin (\frac{59}{50}) (900 x^{2} + 1) = 1

900 x^{2} - sin (\frac{59}{50}) (900 x^{2} + 1) = 1

Expand

- sin (\frac{59}{50}) (900 x^{2} + 1) : - 900 sin (\frac{59}{50}) x^{2} - sin (\frac{59}{50})

- sin (\frac{59}{50}) (900 x^{2} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - sin (\frac{59}{50}), b = 900 x^{2}, c = 1

= - sin (\frac{59}{50}) \cdot 900 x^{2} + (- sin (\frac{59}{50})) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 900 sin (\frac{59}{50}) x^{2} - 1 \cdot sin (\frac{59}{50})

Multiply:

1 \cdot sin (\frac{59}{50}) = sin (\frac{59}{50})

= - 900 sin (\frac{59}{50}) x^{2} - sin (\frac{59}{50})

900 x^{2} - 900 sin (\frac{59}{50}) x^{2} - sin (\frac{59}{50}) = 1

Move

sin (\frac{59}{50})

to the right side

900 x^{2} - 900 sin (\frac{59}{50}) x^{2} - sin (\frac{59}{50}) = 1

Add

sin (\frac{59}{50})

to both sides

900 x^{2} - 900 sin (\frac{59}{50}) x^{2} - sin (\frac{59}{50}) + sin (\frac{59}{50}) = 1 + sin (\frac{59}{50})

Simplify

900 x^{2} - 900 sin (\frac{59}{50}) x^{2} = 1 + sin (\frac{59}{50})

900 x^{2} - 900 sin (\frac{59}{50}) x^{2} = 1 + sin (\frac{59}{50})

Factor

900 x^{2} - 900 sin (\frac{59}{50}) x^{2} : 900 (1 - sin (\frac{59}{50})) x^{2}

900 x^{2} - 900 sin (\frac{59}{50}) x^{2}

Rewrite as

= 1 \cdot 900 x^{2} - 900 x^{2} sin (\frac{59}{50})

Factor out common term

900 x^{2}

= 900 x^{2} (1 - sin (\frac{59}{50}))

900 (1 - sin (\frac{59}{50})) x^{2} = 1 + sin (\frac{59}{50})

Divide both sides by

900 (1 - sin (\frac{59}{50}))

900 (1 - sin (\frac{59}{50})) x^{2} = 1 + sin (\frac{59}{50})

Divide both sides by

900 (1 - sin (\frac{59}{50}))

\frac{900 ( 1 - sin ( \frac{59}{50} ) ) x ^{2}}{900 ( 1 - sin ( \frac{59}{50} ) )} = \frac{1}{900 ( 1 - sin ( \frac{59}{50} ) )} + \frac{sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

Simplify

\frac{900 ( 1 - sin ( \frac{59}{50} ) ) x ^{2}}{900 ( 1 - sin ( \frac{59}{50} ) )} = \frac{1}{900 ( 1 - sin ( \frac{59}{50} ) )} + \frac{sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

Simplify

\frac{900 ( 1 - sin ( \frac{59}{50} ) ) x ^{2}}{900 ( 1 - sin ( \frac{59}{50} ) )} : x^{2}

\frac{900 ( 1 - sin ( \frac{59}{50} ) ) x ^{2}}{900 ( 1 - sin ( \frac{59}{50} ) )}

Divide the numbers:

\frac{900}{900} = 1

= \frac{( - sin ( \frac{59}{50} ) + 1 ) x ^{2}}{1 - sin ( \frac{59}{50} )}

Cancel the common factor:

1 - sin (\frac{59}{50})

= x^{2}

Simplify

\frac{1}{900 ( 1 - sin ( \frac{59}{50} ) )} + \frac{sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )} : \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

\frac{1}{900 ( 1 - sin ( \frac{59}{50} ) )} + \frac{sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

x^{2} = \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

x^{2} = \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

x^{2} = \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}, x = - \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

\frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )} = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

\frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1 + sin ( \frac{59}{50} )}{900 ( - sin ( \frac{59}{50} ) + 1 )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

900 (- sin (\frac{59}{50}) + 1) = 900 - sin (\frac{59}{50}) + 1

= \frac{1 + sin ( \frac{59}{50} )}{900 - sin ( \frac{59}{50} ) + 1}

900 = 30

900

Factor the number:

900 = 3 0^{2}

= 3 0^{2}

Apply radical rule:

3 0^{2} = 30

= 30

= \frac{1 + sin ( \frac{59}{50} )}{30 - sin ( \frac{59}{50} ) + 1}

= \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

- \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )} = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

- \frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

Simplify

\frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )} : \frac{1 + sin ( \frac{59}{50} )}{30 - sin ( \frac{59}{50} ) + 1}

\frac{1 + sin ( \frac{59}{50} )}{900 ( 1 - sin ( \frac{59}{50} ) )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1 + sin ( \frac{59}{50} )}{900 ( - sin ( \frac{59}{50} ) + 1 )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

900 (- sin (\frac{59}{50}) + 1) = 900 - sin (\frac{59}{50}) + 1

= \frac{1 + sin ( \frac{59}{50} )}{900 - sin ( \frac{59}{50} ) + 1}

900 = 30

900

Factor the number:

900 = 3 0^{2}

= 3 0^{2}

Apply radical rule:

3 0^{2} = 30

= 30

= \frac{1 + sin ( \frac{59}{50} )}{30 - sin ( \frac{59}{50} ) + 1}

= - \frac{sin ( \frac{59}{50} ) + 1}{30 - sin ( \frac{59}{50} ) + 1}

= - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

x = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}, x = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

x = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}, x = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

x = \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}, x = - \frac{1 + sin ( \frac{59}{50} )}{30 1 - sin ( \frac{59}{50} )}

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