What is the general solution for sec(β/4)csc(β/4)=2csc(β/4) ?

The general solution for sec(β/4)csc(β/4)=2csc(β/4) is β=(4pi)/3+8pin,β=(20pi)/3+8pin

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sec(β/4)csc(β/4)=2csc(β/4)

Solution

sec (\frac{β}{4}) csc (\frac{β}{4}) = 2 csc (\frac{β}{4})

Solution

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

Degrees

β = 24 0^{\circ} + 144 0^{\circ} n, β = 120 0^{\circ} + 144 0^{\circ} n

Solution steps

sec (\frac{β}{4}) csc (\frac{β}{4}) = 2 csc (\frac{β}{4})

Subtract

2 csc (\frac{β}{4})

from both sides

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4}) = 0

Factor

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4}) : csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2)

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4})

Factor out common term

csc (\frac{β}{4})

= csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2)

csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2) = 0

Solving each part separately

csc (\frac{β}{4}) = 0 or sec (\frac{β}{4}) - 2 = 0

csc (\frac{β}{4}) = 0 :

No Solution

csc (\frac{β}{4}) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

sec (\frac{β}{4}) - 2 = 0 : β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

sec (\frac{β}{4}) - 2 = 0

Move

2

to the right side

sec (\frac{β}{4}) - 2 = 0

Add

2

to both sides

sec (\frac{β}{4}) - 2 + 2 = 0 + 2

Simplify

sec (\frac{β}{4}) = 2

sec (\frac{β}{4}) = 2

General solutions for

sec (\frac{β}{4}) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

\frac{β}{4} = \frac{π}{3} + 2 πn, \frac{β}{4} = \frac{5 π}{3} + 2 πn

\frac{β}{4} = \frac{π}{3} + 2 πn, \frac{β}{4} = \frac{5 π}{3} + 2 πn

Solve

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

\frac{β}{4} = \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{β}{4} = \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{4 β}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 β}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 β}{4} : β

\frac{4 β}{4}

Divide the numbers:

\frac{4}{4} = 1

= β

Simplify

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

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

Solve

\frac{β}{4} = \frac{5 π}{3} + 2 πn : β = \frac{20 π}{3} + 8 πn

\frac{β}{4} = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

\frac{β}{4} = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

\frac{4 β}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 β}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 β}{4} : β

\frac{4 β}{4}

Divide the numbers:

\frac{4}{4} = 1

= β

Simplify

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 4}{3}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

Combine all the solutions

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

Popular Examples

sin(x)= 1/7

sin (x) = \frac{1}{7}

tan(θ/2)=-1

tan (\frac{θ}{2}) = - 1

tan(θ/2)=1

tan (\frac{θ}{2}) = 1

4sin^2(x)cot(x)-2cot(x)=0

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

cos(2θ)+sin(θ)=0

cos (2 θ) + sin (θ) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sec(β/4)csc(β/4)=2csc(β/4) ?
The general solution for sec(β/4)csc(β/4)=2csc(β/4) is β=(4pi)/3+8pin,β=(20pi)/3+8pin