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

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

- 3 csc (x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

2 \cdot \frac{1}{csc ( x )} = \frac{2}{csc ( x )}

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

Multiply fractions:

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

= \frac{1 \cdot 2}{csc ( x )}

Multiply the numbers:

1 \cdot 2 = 2

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

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

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

Solve by substitution

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

Let:

csc (x) = u

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

- 2 + \frac{2}{u} - 3 + u 3 = 0 : u = \frac{2 3}{3}, u = 1

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

\frac{2}{u} u : 2

\frac{2}{u} u

Multiply fractions:

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

= \frac{2 u}{u}

Cancel the common factor:

u

= 2

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

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

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

Solve

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 3, b = - 2 - 3, c = 2

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

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

(- 2 - 3)^{2} - 43 \cdot 2 = 2 - 3

(- 2 - 3)^{2} - 43 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

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

Expand

(- 2 - 3)^{2} - 83 : 7 - 43

(- 2 - 3)^{2} - 83

(- 2 - 3)^{2} : 7 + 43

Apply Perfect Square Formula:

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

a = - 2, b = 3

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

Simplify

(- 2)^{2} - 2 (- 2) 3 + (3)^{2} : 7 + 43

(- 2)^{2} - 2 (- 2) 3 + (3)^{2}

Apply rule

- (- a) = a

= (- 2)^{2} + 2 \cdot 23 + (3)^{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 23 = 43

2 \cdot 23

Multiply the numbers:

2 \cdot 2 = 4

= 43

(3)^{2} = 3

(3)^{2}

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

= 4 + 43 + 3

Add the numbers:

4 + 3 = 7

= 7 + 43

= 7 + 43

= 7 + 43 - 83

Add similar elements:

43 - 83 = - 43

= 7 - 43

= 7 - 43

= 3 - 43 + 4

= (3)^{2} - 43 + (4)^{2}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= (3)^{2} - 43 + 2^{2}

23 \cdot 2 = 43

23 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 43

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

Apply Perfect Square Formula:

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

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

= (3 - 2)^{2}

Apply exponent rule:

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

n

is even

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

= (2 - 3)^{2}

Apply radical rule:

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

= 2 - 3

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

Separate the solutions

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

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

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

Expand

- (- 2 - 3) + 2 - 3 : 4

- (- 2 - 3) + 2 - 3

- (- 2 - 3) : 2 + 3

- (- 2 - 3)

Distribute parentheses

= - (- 2) - (- 3)

Apply minus-plus rules

- (- a) = a

= 2 + 3

= 2 + 3 + 2 - 3

Simplify

2 + 3 + 2 - 3 : 4

2 + 3 + 2 - 3

Add similar elements:

3 - 3 = 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 - 3 ) - ( 2 - 3 )}{2 3} : 1

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

Expand

- (- 2 - 3) - (2 - 3) : 23

- (- 2 - 3) - (2 - 3)

- (- 2 - 3) : 2 + 3

- (- 2 - 3)

Distribute parentheses

= - (- 2) - (- 3)

Apply minus-plus rules

- (- a) = a

= 2 + 3

= 2 + 3 - (2 - 3)

- (2 - 3) : - 2 + 3

- (2 - 3)

Distribute parentheses

= - (2) - (- 3)

Apply minus-plus rules

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

= - 2 + 3

= 2 + 3 - 2 + 3

Simplify

2 + 3 - 2 + 3 : 23

2 + 3 - 2 + 3

Add similar elements:

3 + 3 = 23

= 2 + 23 - 2

2 - 2 = 0

= 23

= 23

= \frac{2 3}{2 3}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 + \frac{2}{u} - 3 + 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 = 1

Substitute back

u = csc (x)

csc (x) = \frac{2 3}{3}, csc (x) = 1

csc (x) = \frac{2 3}{3}, csc (x) = 1

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

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

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

General solutions for

csc (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

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