What is the general solution for tan(x/2)=1 ?

The general solution for tan(x/2)=1 is x= pi/2+2pin

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tan(x/2)=1

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

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

General solutions for

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

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{π}{4} + πn

\frac{x}{2} = \frac{π}{4} + πn

Solve

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

\frac{x}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + πn

Multiply both sides by

2

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

Simplify

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

2 \cdot \frac{π}{4} + 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (\frac{x}{2}) = 0

sin (x) - 1 = cos (x)

cos (2 x) = - cos (x)

sin (x) = 3

- 2 sin (2 x) = 0

What is the general solution for tan(x/2)=1 ?
The general solution for tan(x/2)=1 is x= pi/2+2pin