What is the general solution for (sqrt(2))/2 csc(x)-1=0 ?

The general solution for (sqrt(2))/2 csc(x)-1=0 is x= pi/4+2pin,x=(3pi)/4+2pin

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(sqrt(2))/2 csc(x)-1=0

Solution

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

Solution

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

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Multiply

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

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

Multiply fractions:

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

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

2 csc (x) = 2

2 csc (x) = 2

Divide both sides by

2

2 csc (x) = 2

Divide both sides by

2

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

Simplify

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

Simplify

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

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

Cancel the common factor:

2

= csc (x)

Simplify

\frac{2}{2} : 2

\frac{2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2

csc (x) = 2

csc (x) = 2

csc (x) = 2

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

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cot (x) = \frac{3}{3}

sin^2(x)=1,0<= x<= 2pi

sin^{2} (x) = 1, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for (sqrt(2))/2 csc(x)-1=0 ?
The general solution for (sqrt(2))/2 csc(x)-1=0 is x= pi/4+2pin,x=(3pi)/4+2pin