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4sin(x)=sqrt(3)csc(x)+2-2sqrt(3)

Solution

4 sin (x) = 3 csc (x) + 2 - 23

Solution

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (x) = 3 csc (x) + 2 - 23

Subtract

3 csc (x) + 2 - 23

from both sides

4 sin (x) - 3 csc (x) - 2 + 23 = 0

Rewrite using trig identities

- 2 + 23 + 4 sin (x) - csc (x) 3

Use the basic trigonometric identity:

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

= - 2 + 23 + 4 \cdot \frac{1}{csc ( x )} - csc (x) 3

4 \cdot \frac{1}{csc ( x )} = \frac{4}{csc ( x )}

4 \cdot \frac{1}{csc ( x )}

Multiply fractions:

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

= \frac{1 \cdot 4}{csc ( x )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{csc ( x )}

= - 2 + 23 + \frac{4}{csc ( x )} - 3 csc (x)

- 2 + \frac{4}{csc ( x )} + 23 - csc (x) 3 = 0

Solve by substitution

- 2 + \frac{4}{csc ( x )} + 23 - csc (x) 3 = 0

Let:

csc (x) = u

- 2 + \frac{4}{u} + 23 - u 3 = 0

- 2 + \frac{4}{u} + 23 - u 3 = 0 : u = - \frac{2 3}{3}, u = 2

- 2 + \frac{4}{u} + 23 - u 3 = 0

Multiply both sides by

u

- 2 + \frac{4}{u} + 23 - u 3 = 0

Multiply both sides by

u

- 2 u + \frac{4}{u} u + 23 u - u 3 u = 0 \cdot u

Simplify

- 2 u + \frac{4}{u} u + 23 u - u 3 u = 0 \cdot u

Simplify

\frac{4}{u} u : 4

\frac{4}{u} u

Multiply fractions:

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

= \frac{4 u}{u}

Cancel the common factor:

u

= 4

Simplify

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

- u 3 u

Apply exponent rule:

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

uu = u^{1 + 1}

= - 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 2 u + 4 + 23 u - 3 u^{2} = 0

- 2 u + 4 + 23 u - 3 u^{2} = 0

- 2 u + 4 + 23 u - 3 u^{2} = 0

Solve

- 2 u + 4 + 23 u - 3 u^{2} = 0 : u = - \frac{2 3}{3}, u = 2

- 2 u + 4 + 23 u - 3 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 3 u^{2} + (- 2 + 23) u + 4 = 0

Solve with the quadratic formula

- 3 u^{2} + (- 2 + 23) u + 4 = 0

Quadratic Equation Formula:

For

a = - 3, b = - 2 + 23, c = 4

u_{1, 2} = \frac{- ( - 2 + 2 3 ) \pm ( - 2 + 2 3 ) ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

u_{1, 2} = \frac{- ( - 2 + 2 3 ) \pm ( - 2 + 2 3 ) ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

(- 2 + 23)^{2} - 4 (- 3) \cdot 4 = 23 + 2

(- 2 + 23)^{2} - 4 (- 3) \cdot 4

Apply rule

- (- a) = a

= (- 2 + 23)^{2} + 43 \cdot 4

Multiply the numbers:

4 \cdot 4 = 16

= (23 - 2)^{2} + 163

Expand

(- 2 + 23)^{2} + 163 : 16 + 83

(- 2 + 23)^{2} + 163

(- 2 + 23)^{2} : 16 - 83

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - 2, b = 23

= (- 2)^{2} + 2 (- 2) \cdot 23 + (23)^{2}

Simplify

(- 2)^{2} + 2 (- 2) \cdot 23 + (23)^{2} : 16 - 83

(- 2)^{2} + 2 (- 2) \cdot 23 + (23)^{2}

Remove parentheses:

(- a) = - a

= (- 2)^{2} - 2 \cdot 2 \cdot 23 + (23)^{2}

(- 2)^{2} = 4

(- 2)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

= 2^{2}

2^{2} = 4

= 4

2 \cdot 2 \cdot 23 = 83

2 \cdot 2 \cdot 23

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 83

(23)^{2} = 12

(23)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (3)^{2}

(3)^{2} : 3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 2^{2} \cdot 3

2^{2} = 4

= 4 \cdot 3

Multiply the numbers:

4 \cdot 3 = 12

= 12

= 4 - 83 + 12

Add the numbers:

4 + 12 = 16

= 16 - 83

= 16 - 83

= 16 - 83 + 163

Add similar elements:

- 83 + 163 = 83

= 16 + 83

= 16 + 83

= 12 + 83 + 4

= 4 \cdot 3 + 83 + 4

= (4)^{2} (3)^{2} + 83 + (4)^{2}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= 2^{2} (3)^{2} + 83 + 2^{2}

2 \cdot 23 \cdot 2 = 83

2 \cdot 23 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 83

= (23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2} = (23 + 2)^{2}

= (23 + 2)^{2}

Apply radical rule:

(23 + 2)^{2} = 23 + 2

= 23 + 2

u_{1, 2} = \frac{- ( - 2 + 2 3 ) \pm ( 2 3 + 2 )}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- ( - 2 + 2 3 ) + 2 3 + 2}{2 ( - 3 )}, u_{2} = \frac{- ( - 2 + 2 3 ) - ( 2 3 + 2 )}{2 ( - 3 )}

u = \frac{- ( - 2 + 2 3 ) + 2 3 + 2}{2 ( - 3 )} : - \frac{2 3}{3}

\frac{- ( - 2 + 2 3 ) + 2 3 + 2}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- ( - 2 + 2 3 ) + 2 3 + 2}{- 2 3}

Apply the fraction rule:

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

= - \frac{- ( - 2 + 2 3 ) + 2 3 + 2}{2 3}

Expand

- (- 2 + 23) + 23 + 2 : 4

- (- 2 + 23) + 23 + 2

- (- 2 + 23) : 2 - 23

- (- 2 + 23)

Distribute parentheses

= - (- 2) - (23)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 2 - 23

= 2 - 23 + 23 + 2

Simplify

2 - 23 + 23 + 2 : 4

2 - 23 + 23 + 2

Add similar elements:

- 23 + 23 = 0

= 2 + 2

Add the numbers:

2 + 2 = 4

= 4

= 4

= - \frac{4}{2 3}

Divide the numbers:

\frac{4}{2} = 2

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{- ( - 2 + 2 3 ) - ( 2 3 + 2 )}{2 ( - 3 )} : 2

\frac{- ( - 2 + 2 3 ) - ( 2 3 + 2 )}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- ( - 2 + 2 3 ) - ( 2 3 + 2 )}{- 2 3}

Apply the fraction rule:

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

- (- 2 + 23) - (23 + 2) = - ((23 + 2) + (23 - 2))

= \frac{( 2 3 + 2 ) + ( 2 3 - 2 )}{2 3}

Remove parentheses:

(a) = a

= \frac{2 3 + 2 + 2 3 - 2}{2 3}

23 + 2 + 23 - 2 = 43

23 + 2 + 23 - 2

Add similar elements:

23 + 23 = 43

= 43 + 2 - 2

2 - 2 = 0

= 43

= \frac{4 3}{2 3}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2 3}{3}

Cancel the common factor:

3

= 2

The solutions to the quadratic equation are:

u = - \frac{2 3}{3}, u = 2

u = - \frac{2 3}{3}, u = 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 + \frac{4}{u} + 23 - u 3

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - \frac{2 3}{3}, u = 2

Substitute back

u = csc (x)

csc (x) = - \frac{2 3}{3}, csc (x) = 2

csc (x) = - \frac{2 3}{3}, csc (x) = 2

csc (x) = - \frac{2 3}{3} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

csc (x) = - \frac{2 3}{3}

General solutions for

csc (x) = - \frac{2 3}{3}

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

csc (x) = 2

General solutions for

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 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

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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