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

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

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csc(x/2)-1=0

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

x = π + 4 πn

Degrees

x = 18 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

General solutions for

csc (\frac{x}{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{x}{2} = \frac{π}{2} + 2 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

Solve

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

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

4 cos (x) + 4 sin (x) = 26

7 tan^{2} (θ) - 4 tan (θ) = 0, - 9 0^{\circ} \leq θ \leq 9 0^{\circ}

3 - 4 sin^{2} (x) = 2 cos^{2} (x)

2 tan^{2} (α) = 2 tan (α)

cos^{2} (x) - sin^{2} (x) = - \frac{3}{4}

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