What is the general solution for 0.85=(1-sin(x))/(1+sin(x)) ?

The general solution for 0.85=(1-sin(x))/(1+sin(x)) is x=0.08117…+2pin,x=pi-0.08117…+2pin

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0.85=(1-sin(x))/(1+sin(x))

Solution

0.85 = \frac{1 - sin ( x )}{1 + sin ( x )}

Solution

x = 0.08117 \dots + 2 πn, x = π - 0.08117 \dots + 2 πn

Degrees

x = 4.65070 \dots^{\circ} + 36 0^{\circ} n, x = 175.34929 \dots^{\circ} + 36 0^{\circ} n

Solution steps

0.85 = \frac{1 - sin ( x )}{1 + sin ( x )}

Switch sides

\frac{1 - sin ( x )}{1 + sin ( x )} = 0.85

Solve by substitution

\frac{1 - sin ( x )}{1 + sin ( x )} = 0.85

Let:

sin (x) = u

\frac{1 - u}{1 + u} = 0.85

\frac{1 - u}{1 + u} = 0.85 : u = \frac{3}{37}

\frac{1 - u}{1 + u} = 0.85

Multiply both sides by

1 + u

\frac{1 - u}{1 + u} = 0.85

Multiply both sides by

1 + u

\frac{1 - u}{1 + u} (1 + u) = 0.85 (1 + u)

Simplify

1 - u = 0.85 (1 + u)

1 - u = 0.85 (1 + u)

Multiply both sides by

100

1 - u = 0.85 (1 + u)

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 \cdot 100 - u \cdot 100 = 0.85 (1 + u) \cdot 100

Refine

100 - 100 u = 85 (1 + u)

100 - 100 u = 85 (1 + u)

Expand

85 (1 + u) : 85 + 85 u

85 (1 + u)

Apply the distributive law:

a (b + c) = ab + a c

a = 85, b = 1, c = u

= 85 \cdot 1 + 85 u

Multiply the numbers:

85 \cdot 1 = 85

= 85 + 85 u

100 - 100 u = 85 + 85 u

Move

100

to the right side

100 - 100 u = 85 + 85 u

Subtract

100

from both sides

100 - 100 u - 100 = 85 + 85 u - 100

Simplify

- 100 u = 85 u - 15

- 100 u = 85 u - 15

Move

85 u

to the left side

- 100 u = 85 u - 15

Subtract

85 u

from both sides

- 100 u - 85 u = 85 u - 15 - 85 u

Simplify

- 185 u = - 15

- 185 u = - 15

Divide both sides by

- 185

- 185 u = - 15

Divide both sides by

- 185

\frac{- 185 u}{- 185} = \frac{- 15}{- 185}

Simplify

\frac{- 185 u}{- 185} = \frac{- 15}{- 185}

Simplify

\frac{- 185 u}{- 185} : u

\frac{- 185 u}{- 185}

Apply the fraction rule:

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

= \frac{185 u}{185}

Divide the numbers:

\frac{185}{185} = 1

= u

Simplify

\frac{- 15}{- 185} : \frac{3}{37}

\frac{- 15}{- 185}

Apply the fraction rule:

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

= \frac{15}{185}

Cancel the common factor:

5

= \frac{3}{37}

u = \frac{3}{37}

u = \frac{3}{37}

u = \frac{3}{37}

Verify Solutions

Find undefined (singularity) points:

u = - 1

Take the denominator(s) of

\frac{1 - u}{1 + u}

and compare to zero

Solve

1 + u = 0 : u = - 1

1 + u = 0

Move

1

to the right side

1 + u = 0

Subtract

1

from both sides

1 + u - 1 = 0 - 1

Simplify

u = - 1

u = - 1

The following points are undefined

u = - 1

Combine undefined points with solutions:

u = \frac{3}{37}

Substitute back

u = sin (x)

sin (x) = \frac{3}{37}

sin (x) = \frac{3}{37}

sin (x) = \frac{3}{37} : x = arcsin (\frac{3}{37}) + 2 πn, x = π - arcsin (\frac{3}{37}) + 2 πn

sin (x) = \frac{3}{37}

Apply trig inverse properties

sin (x) = \frac{3}{37}

General solutions for

sin (x) = \frac{3}{37}

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

x = arcsin (\frac{3}{37}) + 2 πn, x = π - arcsin (\frac{3}{37}) + 2 πn

x = arcsin (\frac{3}{37}) + 2 πn, x = π - arcsin (\frac{3}{37}) + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{37}) + 2 πn, x = π - arcsin (\frac{3}{37}) + 2 πn

Show solutions in decimal form

x = 0.08117 \dots + 2 πn, x = π - 0.08117 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 0.85=(1-sin(x))/(1+sin(x)) ?
The general solution for 0.85=(1-sin(x))/(1+sin(x)) is x=0.08117…+2pin,x=pi-0.08117…+2pin