What is the general solution for 1+cot^2(x)=8sin(x) ?

The general solution for 1+cot^2(x)=8sin(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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1+cot^2(x)=8sin(x)

Solution

1 + cot^{2} (x) = 8 sin (x)

Solution

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

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

1 + cot^{2} (x) = 8 sin (x)

Subtract

8 sin (x)

from both sides

1 + cot^{2} (x) - 8 sin (x) = 0

Rewrite using trig identities

1 + cot^{2} (x) - 8 sin (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= 1 + (\frac{cos ( x )}{sin ( x )})^{2} - 8 sin (x)

Apply exponent rule:

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

= 1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 1 + \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x)

1 + \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x) = 0

Solve by substitution

1 + \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} - 8 sin (x) = 0

Let:

sin (x) = u

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0 : u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0

Multiply both sides by

u^{2}

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u = 0

Multiply both sides by

u^{2}

1 \cdot u^{2} + \frac{1 - u ^{2}}{u ^{2}} u^{2} - 8 u u^{2} = 0 \cdot u^{2}

Simplify

1 \cdot u^{2} + \frac{1 - u ^{2}}{u ^{2}} u^{2} - 8 u u^{2} = 0 \cdot u^{2}

Simplify

1 \cdot u^{2} : u^{2}

1 \cdot u^{2}

Multiply:

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

= u^{2}

Simplify

\frac{1 - u ^{2}}{u ^{2}} u^{2} : 1 - u^{2}

\frac{1 - u ^{2}}{u ^{2}} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 1 - u^{2}

Simplify

- 8 u u^{2} : - 8 u^{3}

- 8 u u^{2}

Apply exponent rule:

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

u u^{2} = u^{1 + 2}

= - 8 u^{1 + 2}

Add the numbers:

1 + 2 = 3

= - 8 u^{3}

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

u^{2} + 1 - u^{2} - 8 u^{3} = 0

Simplify

u^{2} + 1 - u^{2} - 8 u^{3} : - 8 u^{3} + 1

u^{2} + 1 - u^{2} - 8 u^{3}

Group like terms

= - 8 u^{3} + u^{2} - u^{2} + 1

Add similar elements:

u^{2} - u^{2} = 0

= - 8 u^{3} + 1

- 8 u^{3} + 1 = 0

- 8 u^{3} + 1 = 0

- 8 u^{3} + 1 = 0

Solve

- 8 u^{3} + 1 = 0 : u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

- 8 u^{3} + 1 = 0

Move

1

to the right side

- 8 u^{3} + 1 = 0

Subtract

1

from both sides

- 8 u^{3} + 1 - 1 = 0 - 1

Simplify

- 8 u^{3} = - 1

- 8 u^{3} = - 1

Divide both sides by

- 8

- 8 u^{3} = - 1

Divide both sides by

- 8

\frac{- 8 u ^{3}}{- 8} = \frac{- 1}{- 8}

Simplify

u^{3} = \frac{1}{8}

u^{3} = \frac{1}{8}

For

x^{3} = f (a)

the solutions are

Apply radical rule:

assuming

a \geq 0, b \geq 0

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Apply rule

= \frac{1}{2}

Simplify

Apply radical rule:

assuming

a \geq 0, b \geq 0

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Apply rule

= \frac{1}{2}

= \frac{1}{2} \cdot \frac{- 1 + 3 i}{2}

Multiply fractions:

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

= \frac{1 \cdot ( - 1 + 3 i )}{2 \cdot 2}

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

Multiply:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

Remove parentheses:

(- a) = - a

= - 1 + 3 i

= \frac{- 1 + 3 i}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 1 + 3 i}{4}

Rewrite

\frac{- 1 + 3 i}{4}

in standard complex form:

- \frac{1}{4} + \frac{3}{4} i

\frac{- 1 + 3 i}{4}

Apply the fraction rule:

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

\frac{- 1 + 3 i}{4} = - \frac{1}{4} + \frac{3 i}{4}

= - \frac{1}{4} + \frac{3 i}{4}

= - \frac{1}{4} + \frac{3}{4} i

Simplify

Apply radical rule:

assuming

a \geq 0, b \geq 0

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Apply rule

= \frac{1}{2}

= \frac{1}{2} \cdot \frac{- 1 - 3 i}{2}

Multiply fractions:

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

= \frac{1 \cdot ( - 1 - 3 i )}{2 \cdot 2}

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

Multiply:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

Remove parentheses:

(- a) = - a

= - 1 - 3 i

= \frac{- 1 - 3 i}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 1 - 3 i}{4}

Rewrite

\frac{- 1 - 3 i}{4}

in standard complex form:

- \frac{1}{4} - \frac{3}{4} i

\frac{- 1 - 3 i}{4}

Apply the fraction rule:

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

\frac{- 1 - 3 i}{4} = - \frac{1}{4} - \frac{3 i}{4}

= - \frac{1}{4} - \frac{3 i}{4}

= - \frac{1}{4} - \frac{3}{4} i

u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

1 + \frac{1 - u ^{2}}{u ^{2}} - 8 u

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{2}, u = - \frac{1}{4} + i \frac{3}{4}, u = - \frac{1}{4} - i \frac{3}{4}

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{4} + i \frac{3}{4}, sin (x) = - \frac{1}{4} - i \frac{3}{4}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{4} + i \frac{3}{4}, sin (x) = - \frac{1}{4} - i \frac{3}{4}

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

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \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}

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

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

sin (x) = - \frac{1}{4} + i \frac{3}{4} :

No Solution

sin (x) = - \frac{1}{4} + i \frac{3}{4}

No Solution

sin (x) = - \frac{1}{4} - i \frac{3}{4} :

No Solution

sin (x) = - \frac{1}{4} - i \frac{3}{4}

No Solution

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for 1+cot^2(x)=8sin(x) ?
The general solution for 1+cot^2(x)=8sin(x) is x= pi/6+2pin,x=(5pi)/6+2pin