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csc^2(x)-2csc(x)=2-4sin(x),0<= x<= 2pi

Solution

csc^{2} (x) - 2 csc (x) = 2 - 4 sin (x), 0 \leq x \leq 2 π

Solution

x = \frac{π}{6}, x = \frac{5 π}{6}, x = \frac{5 π}{4}, x = \frac{7 π}{4}, x = \frac{π}{4}, x = \frac{3 π}{4}

Degrees

x = 3 0^{\circ}, x = 15 0^{\circ}, x = 22 5^{\circ}, x = 31 5^{\circ}, x = 4 5^{\circ}, x = 13 5^{\circ}

Solution steps

csc^{2} (x) - 2 csc (x) = 2 - 4 sin (x), 0 \leq x \leq 2 π

Subtract

2 - 4 sin (x)

from both sides

csc^{2} (x) - 2 csc (x) - 2 + 4 sin (x) = 0

Rewrite using trig identities

- 2 + csc^{2} (x) - 2 csc (x) + 4 sin (x)

Use the basic trigonometric identity:

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

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

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 + csc^{2} (x) - 2 csc (x) + \frac{4}{csc ( x )}

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

Solve by substitution

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

Let:

csc (x) = u

- 2 + u^{2} + \frac{4}{u} - 2 u = 0

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

- 2 + u^{2} + \frac{4}{u} - 2 u = 0

Multiply both sides by

u

- 2 + u^{2} + \frac{4}{u} - 2 u = 0

Multiply both sides by

u

- 2 u + u^{2} u + \frac{4}{u} u - 2 uu = 0 \cdot u

Simplify

- 2 u + u^{2} u + \frac{4}{u} u - 2 uu = 0 \cdot u

Simplify

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

u^{2} u

Apply exponent rule:

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

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

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

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

- 2 uu : - 2 u^{2}

- 2 uu

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

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

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

Solve

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

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

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

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

Factor

u^{3} - 2 u^{2} - 2 u + 4 : (u - 2) (u + 2) (u - 2)

u^{3} - 2 u^{2} - 2 u + 4

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

Factor out

- 2

from

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

- 2 u + 4

Rewrite

4

2 \cdot 2

= - 2 u + 2 \cdot 2

Factor out common term

- 2

= - 2 (u - 2)

Factor out

u^{2}

from

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

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

Apply exponent rule:

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

u^{3} = u u^{2}

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

Factor out common term

u^{2}

= u^{2} (u - 2)

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

Factor out common term

u - 2

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

Factor

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

u^{2} - 2

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (u + 2) (u - 2)

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

(u - 2) (u + 2) (u - 2) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u - 2 = 0 or u + 2 = 0 or u - 2 = 0

Solve

u - 2 = 0 : u = 2

u - 2 = 0

Move

2

to the right side

u - 2 = 0

Add

2

to both sides

u - 2 + 2 = 0 + 2

Simplify

u = 2

u = 2

Solve

u + 2 = 0 : u = - 2

u + 2 = 0

Move

2

to the right side

u + 2 = 0

Subtract

2

from both sides

u + 2 - 2 = 0 - 2

Simplify

u = - 2

u = - 2

Solve

u - 2 = 0 : u = 2

u - 2 = 0

Move

2

to the right side

u - 2 = 0

Add

2

to both sides

u - 2 + 2 = 0 + 2

Simplify

u = 2

u = 2

The solutions are

u = 2, u = - 2, u = 2

u = 2, u = - 2, u = 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 + u^{2} + \frac{4}{u} - 2 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 2, u = 2

Substitute back

u = csc (x)

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

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

csc (x) = 2, 0 \leq x \leq 2 π : x = \frac{π}{6}, x = \frac{5 π}{6}

csc (x) = 2, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{6}, x = \frac{5 π}{6}

csc (x) = - 2, 0 \leq x \leq 2 π : x = \frac{5 π}{4}, x = \frac{7 π}{4}

csc (x) = - 2, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{5 π}{4}, x = \frac{7 π}{4}

csc (x) = 2, 0 \leq x \leq 2 π : x = \frac{π}{4}, x = \frac{3 π}{4}

csc (x) = 2, 0 \leq x \leq 2 π

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}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{4}, x = \frac{3 π}{4}

Combine all the solutions

x = \frac{π}{6}, x = \frac{5 π}{6}, x = \frac{5 π}{4}, x = \frac{7 π}{4}, x = \frac{π}{4}, x = \frac{3 π}{4}

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