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csc(2x)=2sin(2x),0<= x<= 2pi

Solution

csc (2 x) = 2 sin (2 x), 0 \leq x \leq 2 π

Solution

x = \frac{π}{8}, x = \frac{3 π}{8}, x = \frac{9 π}{8}, x = \frac{11 π}{8}, x = \frac{5 π}{8}, x = \frac{7 π}{8}, x = \frac{13 π}{8}, x = \frac{15 π}{8}

Degrees

x = 22. 5^{\circ}, x = 67. 5^{\circ}, x = 202. 5^{\circ}, x = 247. 5^{\circ}, x = 112. 5^{\circ}, x = 157. 5^{\circ}, x = 292. 5^{\circ}, x = 337. 5^{\circ}

Solution steps

csc (2 x) = 2 sin (2 x), 0 \leq x \leq 2 π

Subtract

2 sin (2 x)

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

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

Solve by substitution

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

Let:

csc (2 x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

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

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 2 = 0

u^{2} - 2 = 0

u^{2} - 2 = 0

Solve

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

u^{2} - 2 = 0

Move

2

to the right side

u^{2} - 2 = 0

Add

2

to both sides

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

Simplify

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 2, u = - 2

u = 2, u = - 2

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - \frac{2}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 2

Substitute back

u = csc (2 x)

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

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

csc (2 x) = 2, 0 \leq x \leq 2 π : x = \frac{π}{8}, x = \frac{3 π}{8}, x = \frac{9 π}{8}, x = \frac{11 π}{8}

csc (2 x) = 2, 0 \leq x \leq 2 π

General solutions for

csc (2 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

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

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

Solve

2 x = \frac{π}{4} + 2 πn : x = \frac{π}{8} + πn

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

Solve

2 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{8} + πn

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2} : \frac{3 π}{8} + πn

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{8}, x = \frac{3 π}{8}, x = \frac{9 π}{8}, x = \frac{11 π}{8}

csc (2 x) = - 2, 0 \leq x \leq 2 π : x = \frac{5 π}{8}, x = \frac{7 π}{8}, x = \frac{13 π}{8}, x = \frac{15 π}{8}

csc (2 x) = - 2, 0 \leq x \leq 2 π

General solutions for

csc (2 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

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

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

Solve

2 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{8} + πn

2 x = \frac{5 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2} : \frac{5 π}{8} + πn

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

Apply the fraction rule:

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

= \frac{5 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

Solve

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{8} + πn

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

Apply the fraction rule:

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

= \frac{7 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{5 π}{8}, x = \frac{7 π}{8}, x = \frac{13 π}{8}, x = \frac{15 π}{8}

Combine all the solutions

x = \frac{π}{8}, x = \frac{3 π}{8}, x = \frac{9 π}{8}, x = \frac{11 π}{8}, x = \frac{5 π}{8}, x = \frac{7 π}{8}, x = \frac{13 π}{8}, x = \frac{15 π}{8}

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