What is the general solution for tan(θ/2+pi/4)=-1 ?

The general solution for tan(θ/2+pi/4)=-1 is θ=2pin+pi

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tan(θ/2+pi/4)=-1

Solution

tan (\frac{θ}{2} + \frac{π}{4}) = - 1

Solution

θ = 2 πn + π

Degrees

θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (\frac{θ}{2} + \frac{π}{4}) = - 1

General solutions for

tan (\frac{θ}{2} + \frac{π}{4}) = - 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{θ}{2} + \frac{π}{4} = \frac{3 π}{4} + πn

\frac{θ}{2} + \frac{π}{4} = \frac{3 π}{4} + πn

Solve

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

\frac{θ}{2} + \frac{π}{4} = \frac{3 π}{4} + πn

Move

\frac{π}{4}

to the right side

\frac{θ}{2} + \frac{π}{4} = \frac{3 π}{4} + πn

Subtract

\frac{π}{4}

from both sides

\frac{θ}{2} + \frac{π}{4} - \frac{π}{4} = \frac{3 π}{4} + πn - \frac{π}{4}

Simplify

\frac{θ}{2} + \frac{π}{4} - \frac{π}{4} = \frac{3 π}{4} + πn - \frac{π}{4}

Simplify

\frac{θ}{2} + \frac{π}{4} - \frac{π}{4} : \frac{θ}{2}

\frac{θ}{2} + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= \frac{θ}{2}

Simplify

\frac{3 π}{4} + πn - \frac{π}{4} : πn + \frac{π}{2}

\frac{3 π}{4} + πn - \frac{π}{4}

Group like terms

= πn - \frac{π}{4} + \frac{3 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{3 π}{4} : \frac{π}{2}

Apply rule

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

= \frac{- π + 3 π}{4}

Add similar elements:

- π + 3 π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= πn + \frac{π}{2}

\frac{θ}{2} = πn + \frac{π}{2}

\frac{θ}{2} = πn + \frac{π}{2}

\frac{θ}{2} = πn + \frac{π}{2}

Multiply both sides by

2

\frac{θ}{2} = πn + \frac{π}{2}

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 πn + 2 \cdot \frac{π}{2} : 2 πn + π

2 πn + 2 \cdot \frac{π}{2}

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 πn + π

θ = 2 πn + π

θ = 2 πn + π

θ = 2 πn + π

θ = 2 πn + π

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Frequently Asked Questions (FAQ)

What is the general solution for tan(θ/2+pi/4)=-1 ?
The general solution for tan(θ/2+pi/4)=-1 is θ=2pin+pi