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Popular >

(300)/(1-sin(θ))=400+400sin(θ)

Solution

\frac{300}{1 - sin ( θ )} = 400 + 400 sin (θ)

Solution

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Degrees

θ = 21 0^{\circ} + 36 0^{\circ} n, θ = 33 0^{\circ} + 36 0^{\circ} n, θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{300}{1 - sin ( θ )} = 400 + 400 sin (θ)

Solve by substitution

\frac{300}{1 - sin ( θ )} = 400 + 400 sin (θ)

Let:

sin (θ) = u

\frac{300}{1 - u} = 400 + 400 u

\frac{300}{1 - u} = 400 + 400 u : u = - \frac{1}{2}, u = \frac{1}{2}

\frac{300}{1 - u} = 400 + 400 u

Multiply both sides by

1 - u

\frac{300}{1 - u} = 400 + 400 u

Multiply both sides by

1 - u

\frac{300}{1 - u} (1 - u) = 400 (1 - u) + 400 u (1 - u)

Simplify

300 = 400 (1 - u) + 400 u (1 - u)

300 = 400 (1 - u) + 400 u (1 - u)

Solve

300 = 400 (1 - u) + 400 u (1 - u) : u = - \frac{1}{2}, u = \frac{1}{2}

300 = 400 (1 - u) + 400 u (1 - u)

Expand

400 (1 - u) + 400 u (1 - u) : 400 - 400 u^{2}

400 (1 - u) + 400 u (1 - u)

Expand

400 (1 - u) : 400 - 400 u

400 (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 400, b = 1, c = u

= 400 \cdot 1 - 400 u

Multiply the numbers:

400 \cdot 1 = 400

= 400 - 400 u

= 400 - 400 u + 400 u (1 - u)

Expand

400 u (1 - u) : 400 u - 400 u^{2}

400 u (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 400 u, b = 1, c = u

= 400 u \cdot 1 - 400 uu

= 400 \cdot 1 \cdot u - 400 uu

Simplify

400 \cdot 1 \cdot u - 400 uu : 400 u - 400 u^{2}

400 \cdot 1 \cdot u - 400 uu

400 \cdot 1 \cdot u = 400 u

400 \cdot 1 \cdot u

Multiply the numbers:

400 \cdot 1 = 400

= 400 u

400 uu = 400 u^{2}

400 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 400 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 400 u^{2}

= 400 u - 400 u^{2}

= 400 u - 400 u^{2}

= 400 - 400 u + 400 u - 400 u^{2}

Add similar elements:

- 400 u + 400 u = 0

= 400 - 400 u^{2}

300 = 400 - 400 u^{2}

Switch sides

400 - 400 u^{2} = 300

Move

300

to the left side

400 - 400 u^{2} = 300

Subtract

300

from both sides

400 - 400 u^{2} - 300 = 300 - 300

Simplify

- 400 u^{2} + 100 = 0

- 400 u^{2} + 100 = 0

Solve with the quadratic formula

- 400 u^{2} + 100 = 0

Quadratic Equation Formula:

For

a = - 400, b = 0, c = 100

u_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 ( - 400 ) \cdot 100}{2 ( - 400 )}

u_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 ( - 400 ) \cdot 100}{2 ( - 400 )}

0^{2} - 4 (- 400) \cdot 100 = 400

0^{2} - 4 (- 400) \cdot 100

Apply rule

0^{a} = 0

0^{2} = 0

= 0 - 4 (- 400) \cdot 100

Apply rule

- (- a) = a

= 0 + 4 \cdot 400 \cdot 100

Multiply the numbers:

4 \cdot 400 \cdot 100 = 160000

= 0 + 160000

Add the numbers:

0 + 160000 = 160000

= 160000

Factor the number:

160000 = 40 0^{2}

= 40 0^{2}

Apply radical rule:

40 0^{2} = 400

= 400

u_{1, 2} = \frac{- 0 \pm 400}{2 ( - 400 )}

Separate the solutions

u_{1} = \frac{- 0 + 400}{2 ( - 400 )}, u_{2} = \frac{- 0 - 400}{2 ( - 400 )}

u = \frac{- 0 + 400}{2 ( - 400 )} : - \frac{1}{2}

\frac{- 0 + 400}{2 ( - 400 )}

Remove parentheses:

(- a) = - a

= \frac{- 0 + 400}{- 2 \cdot 400}

Add/Subtract the numbers:

- 0 + 400 = 400

= \frac{400}{- 2 \cdot 400}

Multiply the numbers:

2 \cdot 400 = 800

= \frac{400}{- 800}

Apply the fraction rule:

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

= - \frac{400}{800}

Cancel the common factor:

400

= - \frac{1}{2}

u = \frac{- 0 - 400}{2 ( - 400 )} : \frac{1}{2}

\frac{- 0 - 400}{2 ( - 400 )}

Remove parentheses:

(- a) = - a

= \frac{- 0 - 400}{- 2 \cdot 400}

Subtract the numbers:

- 0 - 400 = - 400

= \frac{- 400}{- 2 \cdot 400}

Multiply the numbers:

2 \cdot 400 = 800

= \frac{- 400}{- 800}

Apply the fraction rule:

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

= \frac{400}{800}

Cancel the common factor:

400

= \frac{1}{2}

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = \frac{1}{2}

u = - \frac{1}{2}, u = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 1

Take the denominator(s) of

\frac{300}{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

Divide both sides by

- 1

- u = - 1

Divide both sides by

- 1

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

Simplify

u = 1

u = 1

The following points are undefined

u = 1

Combine undefined points with solutions:

u = - \frac{1}{2}, u = \frac{1}{2}

Substitute back

u = sin (θ)

sin (θ) = - \frac{1}{2}, sin (θ) = \frac{1}{2}

sin (θ) = - \frac{1}{2}, sin (θ) = \frac{1}{2}

sin (θ) = - \frac{1}{2} : θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

sin (θ) = - \frac{1}{2}

General solutions for

sin (θ) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = \frac{1}{2}

General solutions for

sin (θ) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Combine all the solutions

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

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