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Popular >

(((1-sin^2(x)))/((1+2*sin^2(x))))=0.4

Solution

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 \cdot sin ^{2} ( x ) )}) = 0.4

Solution

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn, x = - 0.61547 \dots + 2 πn, x = π + 0.61547 \dots + 2 πn

Degrees

x = 35.26438 \dots^{\circ} + 36 0^{\circ} n, x = 144.73561 \dots^{\circ} + 36 0^{\circ} n, x = - 35.26438 \dots^{\circ} + 36 0^{\circ} n, x = 215.26438 \dots^{\circ} + 36 0^{\circ} n

Solution steps

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 sin ^{2} ( x ) )}) = 0.4

Solve by substitution

\frac{1 - sin ^{2} ( x )}{1 + 2 sin ^{2} ( x )} = 0.4

Let:

sin (x) = u

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.4

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.4 : u = \frac{1}{3}, u = - \frac{1}{3}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.4

Multiply both sides by

1 + 2 u^{2}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.4

Multiply both sides by

1 + 2 u^{2}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} (1 + 2 u^{2}) = 0.4 (1 + 2 u^{2})

Simplify

1 - u^{2} = 0.4 (1 + 2 u^{2})

1 - u^{2} = 0.4 (1 + 2 u^{2})

Solve

1 - u^{2} = 0.4 (1 + 2 u^{2}) : u = \frac{1}{3}, u = - \frac{1}{3}

1 - u^{2} = 0.4 (1 + 2 u^{2})

Expand

0.4 (1 + 2 u^{2}) : 0.4 + 0.8 u^{2}

0.4 (1 + 2 u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 0.4, b = 1, c = 2 u^{2}

= 0.4 \cdot 1 + 0.4 \cdot 2 u^{2}

= 1 \cdot 0.4 + 2 \cdot 0.4 u^{2}

Simplify

1 \cdot 0.4 + 2 \cdot 0.4 u^{2} : 0.4 + 0.8 u^{2}

1 \cdot 0.4 + 2 \cdot 0.4 u^{2}

Multiply the numbers:

1 \cdot 0.4 = 0.4

= 0.4 + 2 \cdot 0.4 u^{2}

Multiply the numbers:

2 \cdot 0.4 = 0.8

= 0.4 + 0.8 u^{2}

= 0.4 + 0.8 u^{2}

1 - u^{2} = 0.4 + 0.8 u^{2}

Move

1

to the right side

1 - u^{2} = 0.4 + 0.8 u^{2}

Subtract

1

from both sides

1 - u^{2} - 1 = 0.4 + 0.8 u^{2} - 1

Simplify

- u^{2} = 0.8 u^{2} - 0.6

- u^{2} = 0.8 u^{2} - 0.6

Move

0.8 u^{2}

to the left side

- u^{2} = 0.8 u^{2} - 0.6

Subtract

0.8 u^{2}

from both sides

- u^{2} - 0.8 u^{2} = 0.8 u^{2} - 0.6 - 0.8 u^{2}

Simplify

- 1.8 u^{2} = - 0.6

- 1.8 u^{2} = - 0.6

Multiply both sides by

10

- 1.8 u^{2} = - 0.6

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

- 1.8 u^{2} \cdot 10 = - 0.6 \cdot 10

Refine

- 18 u^{2} = - 6

- 18 u^{2} = - 6

Divide both sides by

- 18

- 18 u^{2} = - 6

Divide both sides by

- 18

\frac{- 18 u ^{2}}{- 18} = \frac{- 6}{- 18}

Simplify

\frac{- 18 u ^{2}}{- 18} = \frac{- 6}{- 18}

Simplify

\frac{- 18 u ^{2}}{- 18} : u^{2}

\frac{- 18 u ^{2}}{- 18}

Apply the fraction rule:

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

= \frac{18 u ^{2}}{18}

Divide the numbers:

\frac{18}{18} = 1

= u^{2}

Simplify

\frac{- 6}{- 18} : \frac{1}{3}

\frac{- 6}{- 18}

Apply the fraction rule:

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

= \frac{6}{18}

Cancel the common factor:

6

= \frac{1}{3}

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{3}, u = - \frac{1}{3}

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = sin (x)

sin (x) = \frac{1}{3}, sin (x) = - \frac{1}{3}

sin (x) = \frac{1}{3}, sin (x) = - \frac{1}{3}

sin (x) = \frac{1}{3} : x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

sin (x) = \frac{1}{3}

Apply trig inverse properties

sin (x) = \frac{1}{3}

General solutions for

sin (x) = \frac{1}{3}

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

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

sin (x) = - \frac{1}{3} : x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

sin (x) = - \frac{1}{3}

Apply trig inverse properties

sin (x) = - \frac{1}{3}

General solutions for

sin (x) = - \frac{1}{3}

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

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

Combine all the solutions

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn, x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn, x = - 0.61547 \dots + 2 πn, x = π + 0.61547 \dots + 2 πn

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