What is the general solution for csc(θ/2)=sin(θ/2) ?

The general solution for csc(θ/2)=sin(θ/2) is θ=pi+4pin,θ=3pi+4pin

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csc(θ/2)=sin(θ/2)

Solution

csc (\frac{θ}{2}) = sin (\frac{θ}{2})

Solution

θ = π + 4 πn, θ = 3 π + 4 πn

Degrees

θ = 18 0^{\circ} + 72 0^{\circ} n, θ = 54 0^{\circ} + 72 0^{\circ} n

Solution steps

csc (\frac{θ}{2}) = sin (\frac{θ}{2})

Subtract

sin (\frac{θ}{2})

from both sides

csc (\frac{θ}{2}) - sin (\frac{θ}{2}) = 0

Rewrite using trig identities

csc (\frac{θ}{2}) - sin (\frac{θ}{2})

Use the basic trigonometric identity:

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

= csc (\frac{θ}{2}) - \frac{1}{csc ( \frac{θ}{2} )}

csc (\frac{θ}{2}) - \frac{1}{csc ( \frac{θ}{2} )} = 0

Solve by substitution

csc (\frac{θ}{2}) - \frac{1}{csc ( \frac{θ}{2} )} = 0

Let:

csc (\frac{θ}{2}) = u

u - \frac{1}{u} = 0

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

u - \frac{1}{u} = 0

Multiply both sides by

u

u - \frac{1}{u} = 0

Multiply both sides by

u

uu - \frac{1}{u} u = 0 \cdot u

Simplify

uu - \frac{1}{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{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 1 = 0

u^{2} - 1 = 0

u^{2} - 1 = 0

Solve

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

u^{2} - 1 = 0

Move

1

to the right side

u^{2} - 1 = 0

Add

1

to both sides

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - \frac{1}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 1

Substitute back

u = csc (\frac{θ}{2})

csc (\frac{θ}{2}) = 1, csc (\frac{θ}{2}) = - 1

csc (\frac{θ}{2}) = 1, csc (\frac{θ}{2}) = - 1

csc (\frac{θ}{2}) = 1 : θ = π + 4 πn

csc (\frac{θ}{2}) = 1

General solutions for

csc (\frac{θ}{2}) = 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

\frac{θ}{2} = \frac{π}{2} + 2 πn

\frac{θ}{2} = \frac{π}{2} + 2 πn

Solve

\frac{θ}{2} = \frac{π}{2} + 2 πn : θ = π + 4 πn

\frac{θ}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

θ = π + 4 πn

θ = π + 4 πn

θ = π + 4 πn

θ = π + 4 πn

csc (\frac{θ}{2}) = - 1 : θ = 3 π + 4 πn

csc (\frac{θ}{2}) = - 1

General solutions for

csc (\frac{θ}{2}) = - 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

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

Solve

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

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

θ = 3 π + 4 πn

θ = 3 π + 4 πn

θ = 3 π + 4 πn

θ = 3 π + 4 πn

Combine all the solutions

θ = π + 4 πn, θ = 3 π + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc(θ/2)=sin(θ/2) ?
The general solution for csc(θ/2)=sin(θ/2) is θ=pi+4pin,θ=3pi+4pin