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Popular >

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

Solution

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

Solution

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 csc (x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

Solve by substitution

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

Let:

csc (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

- u 2 u

Apply exponent rule:

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

uu = u^{1 + 1}

= - 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 2 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 2, b = - 2 + 2, c = 2

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

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

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

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

Apply rule

- (- a) = a

= (- 2 + 2)^{2} + 42 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= (2 - 2)^{2} + 82

Expand

(- 2 + 2)^{2} + 82 : 6 + 42

(- 2 + 2)^{2} + 82

(- 2 + 2)^{2} : 6 - 42

Apply Perfect Square Formula:

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

a = - 2, b = 2

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

Simplify

(- 2)^{2} + 2 (- 2) 2 + (2)^{2} : 6 - 42

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

Remove parentheses:

(- a) = - a

= (- 2)^{2} - 2 \cdot 22 + (2)^{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 22 = 42

2 \cdot 22

Multiply the numbers:

2 \cdot 2 = 4

= 42

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

= 4 - 42 + 2

Add the numbers:

4 + 2 = 6

= 6 - 42

= 6 - 42

= 6 - 42 + 82

Add similar elements:

- 42 + 82 = 42

= 6 + 42

= 6 + 42

= 2 + 42 + 4

= (2)^{2} + 42 + (4)^{2}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= (2)^{2} + 42 + 2^{2}

22 \cdot 2 = 42

22 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 42

= (2)^{2} + 22 \cdot 2 + 2^{2}

Apply Perfect Square Formula:

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

(2)^{2} + 22 \cdot 2 + 2^{2} = (2 + 2)^{2}

= (2 + 2)^{2}

Apply radical rule:

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

= 2 + 2

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

Separate the solutions

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

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

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

Remove parentheses:

(- a) = - a

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

Apply the fraction rule:

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

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

Expand

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

- (- 2 + 2) + 2 + 2

- (- 2 + 2) : 2 - 2

- (- 2 + 2)

Distribute parentheses

= - (- 2) - (2)

Apply minus-plus rules

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

= 2 - 2

= 2 - 2 + 2 + 2

Simplify

2 - 2 + 2 + 2 : 4

2 - 2 + 2 + 2

Add similar elements:

- 2 + 2 = 0

= 2 + 2

Add the numbers:

2 + 2 = 4

= 4

= 4

= - \frac{4}{2 2}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2}{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= - 2

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

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

Remove parentheses:

(- a) = - a

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

Apply the fraction rule:

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

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

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

Remove parentheses:

(a) = a

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

2 + 2 + 2 - 2 = 22

2 + 2 + 2 - 2

Add similar elements:

2 + 2 = 22

= 2 + 22 - 2

2 - 2 = 0

= 22

= \frac{2 2}{2 2}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - 2, u = 1

u = - 2, u = 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 + \frac{2}{u} + 2 - u 2

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 2, u = 1

Substitute back

u = csc (x)

csc (x) = - 2, csc (x) = 1

csc (x) = - 2, csc (x) = 1

csc (x) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{2} + 2 πn

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