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arcsin(x/(sqrt(3^2+x^2)))=0.01117

Solution

arcsin (\frac{x}{3 ^{2} + x ^{2}}) = 0.01117

Solution

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Solution steps

arcsin (\frac{x}{3 ^{2} + x ^{2}}) = 0.01117

Apply trig inverse properties

arcsin (\frac{x}{3 ^{2} + x ^{2}}) = 0.01117

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

\frac{x}{3 ^{2} + x ^{2}} = sin (0.01117)

sin (0.01117) = sin (\frac{1117}{100000})

sin (0.01117)

\frac{x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000})

\frac{x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000})

Solve

\frac{x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000}) : x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

\frac{x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000})

Multiply both sides by

9 + x^{2}

\frac{x}{3 ^{2} + x ^{2}} 9 + x^{2} = sin (\frac{1117}{100000}) 9 + x^{2}

Simplify

\frac{9 + x ^{2} x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000}) 9 + x^{2}

Square both sides:

x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

\frac{9 + x ^{2} x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000}) 9 + x^{2}

(\frac{9 + x ^{2} x}{3 ^{2} + x ^{2}})^{2} = (sin (\frac{1117}{100000}) 9 + x^{2})^{2}

Expand

(\frac{9 + x ^{2} x}{3 ^{2} + x ^{2}})^{2} : x^{2}

(\frac{9 + x ^{2} x}{3 ^{2} + x ^{2}})^{2}

Apply exponent rule:

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

= \frac{( 9 + x ^{2} x ) ^{2}}{( 3 ^{2} + x ^{2} ) ^{2}}

(3^{2} + x^{2})^{2} : 3^{2} + x^{2}

Apply radical rule:

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

= ((3^{2} + x^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (3^{2} + 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

= 3^{2} + x^{2}

= \frac{( 9 + x ^{2} x ) ^{2}}{3 ^{2} + x ^{2}}

Apply exponent rule:

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

(9 + x^{2} x)^{2} = x^{2} (9 + x^{2})^{2}

= \frac{x ^{2} ( 9 + x ^{2} ) ^{2}}{3 ^{2} + x ^{2}}

(9 + x^{2})^{2} : 9 + x^{2}

Apply radical rule:

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

= ((9 + x^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (9 + 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

= 9 + x^{2}

= \frac{( 9 + x ^{2} ) x ^{2}}{3 ^{2} + x ^{2}}

Expand

\frac{( 9 + x ^{2} ) x ^{2}}{3 ^{2} + x ^{2}} : x^{2}

\frac{( 9 + x ^{2} ) x ^{2}}{3 ^{2} + x ^{2}}

3^{2} = 9

= \frac{x ^{2} ( x ^{2} + 9 )}{9 + x ^{2}}

Cancel the common factor:

9 + x^{2}

= x^{2}

= x^{2}

Expand

(sin (\frac{1117}{100000}) 9 + x^{2})^{2} : 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

(sin (\frac{1117}{100000}) 9 + x^{2})^{2}

Apply exponent rule:

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

= sin^{2} (\frac{1117}{100000}) (9 + x^{2})^{2}

(9 + x^{2})^{2} : 9 + x^{2}

Apply radical rule:

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

= ((9 + x^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (9 + 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

= 9 + x^{2}

= sin^{2} (\frac{1117}{100000}) (9 + x^{2})

Expand

sin^{2} (\frac{1117}{100000}) (9 + x^{2}) : 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

sin^{2} (\frac{1117}{100000}) (9 + x^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = sin^{2} (\frac{1117}{100000}), b = 9, c = x^{2}

= sin^{2} (\frac{1117}{100000}) \cdot 9 + sin^{2} (\frac{1117}{100000}) x^{2}

= 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

= 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

Solve

x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2} : x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}, x = - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

Move

sin^{2} (\frac{1117}{100000}) x^{2}

to the left side

x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2}

Subtract

sin^{2} (\frac{1117}{100000}) x^{2}

from both sides

x^{2} - sin^{2} (\frac{1117}{100000}) x^{2} = 9 sin^{2} (\frac{1117}{100000}) + sin^{2} (\frac{1117}{100000}) x^{2} - sin^{2} (\frac{1117}{100000}) x^{2}

Simplify

x^{2} - sin^{2} (\frac{1117}{100000}) x^{2} = 9 sin^{2} (\frac{1117}{100000})

x^{2} - sin^{2} (\frac{1117}{100000}) x^{2} = 9 sin^{2} (\frac{1117}{100000})

Factor

x^{2} - sin^{2} (\frac{1117}{100000}) x^{2} : (1 - sin^{2} (\frac{1117}{100000})) x^{2}

x^{2} - sin^{2} (\frac{1117}{100000}) x^{2}

Factor out common term

x^{2}

= x^{2} (1 - sin^{2} (\frac{1117}{100000}))

(1 - sin^{2} (\frac{1117}{100000})) x^{2} = 9 sin^{2} (\frac{1117}{100000})

Divide both sides by

1 - sin^{2} (\frac{1117}{100000})

(1 - sin^{2} (\frac{1117}{100000})) x^{2} = 9 sin^{2} (\frac{1117}{100000})

Divide both sides by

1 - sin^{2} (\frac{1117}{100000})

\frac{( 1 - sin ^{2} ( \frac{1117}{100000} ) ) x ^{2}}{1 - sin ^{2} ( \frac{1117}{100000} )} = \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Simplify

x^{2} = \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

x^{2} = \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

For

x^{2} = f (a)

the solutions are

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

x = \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}, x = - \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

\frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

\frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

9 sin^{2} (\frac{1117}{100000}) = 9 sin^{2} (\frac{1117}{100000})

= \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0

sin^{2} (\frac{1117}{100000}) = sin (\frac{1117}{100000})

= \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

- \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} = - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

- \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Simplify

\frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} : \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

\frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

9 sin^{2} (\frac{1117}{100000}) = 9 sin^{2} (\frac{1117}{100000})

= \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0

sin^{2} (\frac{1117}{100000}) = sin (\frac{1117}{100000})

= \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

= - \frac{3 sin ( \frac{1117}{100000} )}{- sin ^{2} ( \frac{1117}{100000} ) + 1}

= - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}, x = - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}, x = - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Verify Solutions:

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

True

, x = - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

False

Check the solutions by plugging them into

\frac{x}{3 ^{2} + x ^{2}} = sin (\frac{1117}{100000})

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

Plug in

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} :

True

\frac{( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} )}{3 ^{2} + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}} = sin (\frac{1117}{100000})

\frac{( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} )}{3 ^{2} + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}} = sin (\frac{1117}{100000})

\frac{\frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )}}{3 ^{2} + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}}

3^{2} = 9

= \frac{\frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )}}{9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{- s i n ^{2} ( \frac{1117}{100000} ) + 1} ) ^{2}}

Apply the fraction rule:

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

= \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} ) 9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}}

9 + \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2} = \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 + \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

\frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2} = \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

\frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

Apply exponent rule:

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

= \frac{( 3 sin ( \frac{1117}{100000} ) ) ^{2}}{( 1 - sin ^{2} ( \frac{1117}{100000} ) ) ^{2}}

(1 - sin^{2} (\frac{1117}{100000}))^{2} : 1 - sin^{2} (\frac{1117}{100000})

Apply radical rule:

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

= ((1 - sin^{2} (\frac{1117}{100000}))^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - sin^{2} (\frac{1117}{100000}))^{\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

= 1 - sin^{2} (\frac{1117}{100000})

= \frac{( 3 sin ( \frac{1117}{100000} ) ) ^{2}}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply exponent rule:

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

(3 sin (\frac{1117}{100000}))^{2} = 3^{2} sin^{2} (\frac{1117}{100000})

= \frac{3 ^{2} sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

3^{2} = 9

= \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

= 9 + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{- sin ^{2} ( \frac{1117}{100000} ) + 1}

Join

9 + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} : \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Convert element to fraction:

9 = \frac{9 ( 1 - sin ^{2} ( \frac{1117}{100000} ) )}{1 - sin ^{2} ( \frac{1117}{100000} )}

= \frac{9 ( 1 - sin ^{2} ( \frac{1117}{100000} ) )}{1 - sin ^{2} ( \frac{1117}{100000} )} + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Since the denominators are equal, combine the fractions:

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

= \frac{9 ( 1 - sin ^{2} ( \frac{1117}{100000} ) ) + 9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Expand

9 (1 - sin^{2} (\frac{1117}{100000})) + 9 sin^{2} (\frac{1117}{100000}) : 9

9 (1 - sin^{2} (\frac{1117}{100000})) + 9 sin^{2} (\frac{1117}{100000})

Expand

9 (1 - sin^{2} (\frac{1117}{100000})) : 9 - 9 sin^{2} (\frac{1117}{100000})

9 (1 - sin^{2} (\frac{1117}{100000}))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = sin^{2} (\frac{1117}{100000})

= 9 \cdot 1 - 9 sin^{2} (\frac{1117}{100000})

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 sin^{2} (\frac{1117}{100000})

= 9 - 9 sin^{2} (\frac{1117}{100000}) + 9 sin^{2} (\frac{1117}{100000})

Add similar elements:

- 9 sin^{2} (\frac{1117}{100000}) + 9 sin^{2} (\frac{1117}{100000}) = 0

= 9

= \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

= \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )}

= \frac{3 sin ( \frac{1117}{100000} )}{\frac{3}{- s i n ^{2} ( \frac{1117}{100000} ) + 1} - sin ^{2} ( \frac{1117}{100000} ) + 1}

Multiply

1 - sin^{2} (\frac{1117}{100000}) \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )} : 3

1 - sin^{2} (\frac{1117}{100000}) \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )}

Multiply fractions:

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

= \frac{3 1 - sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Cancel the common factor:

1 - sin^{2} (\frac{1117}{100000})

= 3

= \frac{3 sin ( \frac{1117}{100000} )}{3}

Divide the numbers:

\frac{3}{3} = 1

= sin (\frac{1117}{100000})

sin (\frac{1117}{100000}) = sin (\frac{1117}{100000})

True

Plug in

x = - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} :

False

\frac{( - \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} )}{3 ^{2} + ( - \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}} = sin (\frac{1117}{100000})

\frac{( - \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} )}{3 ^{2} + ( - \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}} = - sin (\frac{1117}{100000})

\frac{- \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )}}{3 ^{2} + ( - \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}}

3^{2} + - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2} = 9 + \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

3^{2} + - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

3^{2} = 9

= 9 + - \frac{3 sin ( \frac{1117}{100000} )}{- sin ^{2} ( \frac{1117}{100000} ) + 1}^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

- \frac{3 sin ( \frac{1117}{100000} )}{- sin ^{2} ( \frac{1117}{100000} ) + 1}^{2} = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

= 9 + \frac{3 sin ( \frac{1117}{100000} )}{- sin ^{2} ( \frac{1117}{100000} ) + 1}^{2}

= \frac{- \frac{3 s i n ( \frac{1117}{100000} )}{- s i n ^{2} ( \frac{1117}{100000} ) + 1}}{9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{- s i n ^{2} ( \frac{1117}{100000} ) + 1} ) ^{2}}

Apply the fraction rule:

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

= - \frac{\frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )}}{9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}}

Apply the fraction rule:

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

\frac{\frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )}}{9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}} = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} ) 9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}}

= - \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} ) 9 + ( \frac{3 s i n ( \frac{1117}{100000} )}{1 - s i n ^{2} ( \frac{1117}{100000} )} ) ^{2}}

9 + \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2} = \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 + \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

\frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2} = \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

\frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}^{2}

Apply exponent rule:

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

= \frac{( 3 sin ( \frac{1117}{100000} ) ) ^{2}}{( 1 - sin ^{2} ( \frac{1117}{100000} ) ) ^{2}}

(1 - sin^{2} (\frac{1117}{100000}))^{2} : 1 - sin^{2} (\frac{1117}{100000})

Apply radical rule:

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

= ((1 - sin^{2} (\frac{1117}{100000}))^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - sin^{2} (\frac{1117}{100000}))^{\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

= 1 - sin^{2} (\frac{1117}{100000})

= \frac{( 3 sin ( \frac{1117}{100000} ) ) ^{2}}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply exponent rule:

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

(3 sin (\frac{1117}{100000}))^{2} = 3^{2} sin^{2} (\frac{1117}{100000})

= \frac{3 ^{2} sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

3^{2} = 9

= \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

= 9 + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{- sin ^{2} ( \frac{1117}{100000} ) + 1}

Join

9 + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )} : \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Convert element to fraction:

9 = \frac{9 ( 1 - sin ^{2} ( \frac{1117}{100000} ) )}{1 - sin ^{2} ( \frac{1117}{100000} )}

= \frac{9 ( 1 - sin ^{2} ( \frac{1117}{100000} ) )}{1 - sin ^{2} ( \frac{1117}{100000} )} + \frac{9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Since the denominators are equal, combine the fractions:

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

= \frac{9 ( 1 - sin ^{2} ( \frac{1117}{100000} ) ) + 9 sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Expand

9 (1 - sin^{2} (\frac{1117}{100000})) + 9 sin^{2} (\frac{1117}{100000}) : 9

9 (1 - sin^{2} (\frac{1117}{100000})) + 9 sin^{2} (\frac{1117}{100000})

Expand

9 (1 - sin^{2} (\frac{1117}{100000})) : 9 - 9 sin^{2} (\frac{1117}{100000})

9 (1 - sin^{2} (\frac{1117}{100000}))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = sin^{2} (\frac{1117}{100000})

= 9 \cdot 1 - 9 sin^{2} (\frac{1117}{100000})

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 sin^{2} (\frac{1117}{100000})

= 9 - 9 sin^{2} (\frac{1117}{100000}) + 9 sin^{2} (\frac{1117}{100000})

Add similar elements:

- 9 sin^{2} (\frac{1117}{100000}) + 9 sin^{2} (\frac{1117}{100000}) = 0

= 9

= \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

= \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{1 - sin ^{2} ( \frac{1117}{100000} )}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )}

= - \frac{3 sin ( \frac{1117}{100000} )}{\frac{3}{- s i n ^{2} ( \frac{1117}{100000} ) + 1} - sin ^{2} ( \frac{1117}{100000} ) + 1}

Multiply

1 - sin^{2} (\frac{1117}{100000}) \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )} : 3

1 - sin^{2} (\frac{1117}{100000}) \frac{3}{1 - sin ^{2} ( \frac{1117}{100000} )}

Multiply fractions:

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

= \frac{3 1 - sin ^{2} ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

Cancel the common factor:

1 - sin^{2} (\frac{1117}{100000})

= 3

= - \frac{3 sin ( \frac{1117}{100000} )}{3}

Divide the numbers:

\frac{3}{3} = 1

= - sin (\frac{1117}{100000})

- sin (\frac{1117}{100000}) = sin (\frac{1117}{100000})

False

The solution is

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

x = \frac{3 sin ( \frac{1117}{100000} )}{1 - sin ^{2} ( \frac{1117}{100000} )}

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