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Popular >

(((1-sin^2(x)))/((1+2*sin^2(x))))=0.45

Solution

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 \cdot sin ^{2} ( x ) )}) = 0.45

Solution

x = 0.56809 \dots + 2 πn, x = π - 0.56809 \dots + 2 πn, x = - 0.56809 \dots + 2 πn, x = π + 0.56809 \dots + 2 πn

Degrees

x = 32.54946 \dots^{\circ} + 36 0^{\circ} n, x = 147.45053 \dots^{\circ} + 36 0^{\circ} n, x = - 32.54946 \dots^{\circ} + 36 0^{\circ} n, x = 212.54946 \dots^{\circ} + 36 0^{\circ} n

Solution steps

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 sin ^{2} ( x ) )}) = 0.45

Solve by substitution

\frac{1 - sin ^{2} ( x )}{1 + 2 sin ^{2} ( x )} = 0.45

Let:

sin (x) = u

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45 : u = \frac{11}{38}, u = - \frac{11}{38}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45

Multiply both sides by

1 + 2 u^{2}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45

Multiply both sides by

1 + 2 u^{2}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} (1 + 2 u^{2}) = 0.45 (1 + 2 u^{2})

Simplify

1 - u^{2} = 0.45 (1 + 2 u^{2})

1 - u^{2} = 0.45 (1 + 2 u^{2})

Solve

1 - u^{2} = 0.45 (1 + 2 u^{2}) : u = \frac{11}{38}, u = - \frac{11}{38}

1 - u^{2} = 0.45 (1 + 2 u^{2})

Expand

0.45 (1 + 2 u^{2}) : 0.45 + 0.9 u^{2}

0.45 (1 + 2 u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 0.45, b = 1, c = 2 u^{2}

= 0.45 \cdot 1 + 0.45 \cdot 2 u^{2}

= 1 \cdot 0.45 + 2 \cdot 0.45 u^{2}

Simplify

1 \cdot 0.45 + 2 \cdot 0.45 u^{2} : 0.45 + 0.9 u^{2}

1 \cdot 0.45 + 2 \cdot 0.45 u^{2}

Multiply the numbers:

1 \cdot 0.45 = 0.45

= 0.45 + 2 \cdot 0.45 u^{2}

Multiply the numbers:

2 \cdot 0.45 = 0.9

= 0.45 + 0.9 u^{2}

= 0.45 + 0.9 u^{2}

1 - u^{2} = 0.45 + 0.9 u^{2}

Move

1

to the right side

1 - u^{2} = 0.45 + 0.9 u^{2}

Subtract

1

from both sides

1 - u^{2} - 1 = 0.45 + 0.9 u^{2} - 1

Simplify

- u^{2} = 0.9 u^{2} - 0.55

- u^{2} = 0.9 u^{2} - 0.55

Move

0.9 u^{2}

to the left side

- u^{2} = 0.9 u^{2} - 0.55

Subtract

0.9 u^{2}

from both sides

- u^{2} - 0.9 u^{2} = 0.9 u^{2} - 0.55 - 0.9 u^{2}

Simplify

- 1.9 u^{2} = - 0.55

- 1.9 u^{2} = - 0.55

Multiply both sides by

100

- 1.9 u^{2} = - 0.55

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 1.9 u^{2} \cdot 100 = - 0.55 \cdot 100

Refine

- 190 u^{2} = - 55

- 190 u^{2} = - 55

Divide both sides by

- 190

- 190 u^{2} = - 55

Divide both sides by

- 190

\frac{- 190 u ^{2}}{- 190} = \frac{- 55}{- 190}

Simplify

\frac{- 190 u ^{2}}{- 190} = \frac{- 55}{- 190}

Simplify

\frac{- 190 u ^{2}}{- 190} : u^{2}

\frac{- 190 u ^{2}}{- 190}

Apply the fraction rule:

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

= \frac{190 u ^{2}}{190}

Divide the numbers:

\frac{190}{190} = 1

= u^{2}

Simplify

\frac{- 55}{- 190} : \frac{11}{38}

\frac{- 55}{- 190}

Apply the fraction rule:

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

= \frac{55}{190}

Cancel the common factor:

5

= \frac{11}{38}

u^{2} = \frac{11}{38}

u^{2} = \frac{11}{38}

u^{2} = \frac{11}{38}

For

x^{2} = f (a)

the solutions are

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

u = \frac{11}{38}, u = - \frac{11}{38}

u = \frac{11}{38}, u = - \frac{11}{38}

Substitute back

u = sin (x)

sin (x) = \frac{11}{38}, sin (x) = - \frac{11}{38}

sin (x) = \frac{11}{38}, sin (x) = - \frac{11}{38}

sin (x) = \frac{11}{38} : x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn

sin (x) = \frac{11}{38}

Apply trig inverse properties

sin (x) = \frac{11}{38}

General solutions for

sin (x) = \frac{11}{38}

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

x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn

x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn

sin (x) = - \frac{11}{38} : x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

sin (x) = - \frac{11}{38}

Apply trig inverse properties

sin (x) = - \frac{11}{38}

General solutions for

sin (x) = - \frac{11}{38}

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

x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

Combine all the solutions

x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn, x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

Show solutions in decimal form

x = 0.56809 \dots + 2 πn, x = π - 0.56809 \dots + 2 πn, x = - 0.56809 \dots + 2 πn, x = π + 0.56809 \dots + 2 πn

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