What is the general solution for tan(θ/2+pi/3)=1 ?

The general solution for tan(θ/2+pi/3)=1 is θ=2pin-pi/6

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tan(θ/2+pi/3)=1

Solution

tan (\frac{θ}{2} + \frac{π}{3}) = 1

Solution

θ = 2 πn - \frac{π}{6}

Degrees

θ = - 3 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (\frac{θ}{2} + \frac{π}{3}) = 1

General solutions for

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

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

Solve

\frac{θ}{2} + \frac{π}{3} = \frac{π}{4} + πn : θ = 2 πn - \frac{π}{6}

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

Move

\frac{π}{3}

to the right side

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

Simplify

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

\frac{θ}{2} + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= \frac{θ}{2}

Simplify

\frac{π}{4} + πn - \frac{π}{3} : πn - \frac{π}{12}

\frac{π}{4} + πn - \frac{π}{3}

Group like terms

= πn + \frac{π}{4} - \frac{π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

= \frac{π 3}{12} - \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π 4}{12}

Add similar elements:

3 π - 4 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= πn - \frac{π}{12}

\frac{θ}{2} = πn - \frac{π}{12}

\frac{θ}{2} = πn - \frac{π}{12}

\frac{θ}{2} = πn - \frac{π}{12}

Multiply both sides by

2

\frac{θ}{2} = πn - \frac{π}{12}

Multiply both sides by

2

\frac{2 θ}{2} = 2 πn - 2 \cdot \frac{π}{12}

Simplify

\frac{2 θ}{2} = 2 πn - 2 \cdot \frac{π}{12}

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}{12} : 2 πn - \frac{π}{6}

2 πn - 2 \cdot \frac{π}{12}

2 \cdot \frac{π}{12} = \frac{π}{6}

2 \cdot \frac{π}{12}

Multiply fractions:

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

= \frac{π 2}{12}

Cancel the common factor:

2

= \frac{π}{6}

= 2 πn - \frac{π}{6}

θ = 2 πn - \frac{π}{6}

θ = 2 πn - \frac{π}{6}

θ = 2 πn - \frac{π}{6}

θ = 2 πn - \frac{π}{6}

Popular Examples

2sin(piθ)=1

2 sin (π θ) = 1

2sqrt(3)sin(x/2)=3

23 sin (\frac{x}{2}) = 3

2cos(θ)=-1

2 cos (θ) = - 1

sin(θ)= 2/5

sin (θ) = \frac{2}{5}

cos(2x)-cos(x)=0,0<= x<= 2pi

cos (2 x) - cos (x) = 0, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for tan(θ/2+pi/3)=1 ?
The general solution for tan(θ/2+pi/3)=1 is θ=2pin-pi/6