What is the general solution for csc(x/4)=sqrt(2) ?

The general solution for csc(x/4)=sqrt(2) is x=pi+8pin,x=3pi+8pin

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csc(x/4)=sqrt(2)

Solution

csc (\frac{x}{4}) = 2

Solution

x = π + 8 πn, x = 3 π + 8 πn

Degrees

x = 18 0^{\circ} + 144 0^{\circ} n, x = 54 0^{\circ} + 144 0^{\circ} n

Solution steps

csc (\frac{x}{4}) = 2

General solutions for

csc (\frac{x}{4}) = 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

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

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

Solve

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

\frac{x}{4} = \frac{π}{4} + 2 πn

Multiply both sides by

4

\frac{x}{4} = \frac{π}{4} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

4 \cdot \frac{π}{4} + 4 \cdot 2 πn

4 \cdot \frac{π}{4} = π

4 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 4}{4}

Cancel the common factor:

4

= π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= π + 8 πn

x = π + 8 πn

x = π + 8 πn

x = π + 8 πn

Solve

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

\frac{x}{4} = \frac{3 π}{4} + 2 πn

Multiply both sides by

4

\frac{x}{4} = \frac{3 π}{4} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

4 \cdot \frac{3 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{4} = 3 π

4 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 4}{4}

Cancel the common factor:

4

= 3 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 3 π + 8 πn

x = 3 π + 8 πn

x = 3 π + 8 πn

x = 3 π + 8 πn

x = π + 8 πn, x = 3 π + 8 πn

Popular Examples

4cos(θ)=sec(θ)

4 cos (θ) = sec (θ)

sin(x)+sqrt(3)cos(x)=-sqrt(2)

sin (x) + 3 cos (x) = - 2

cot(θ)+4csc(θ)=6

cot (θ) + 4 csc (θ) = 6

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

sin (x) 2 = - sin (x)

tan^3(x)=1

tan^{3} (x) = 1

Frequently Asked Questions (FAQ)

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