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

The general solution for cos(θ/2+pi/2)= 1/2 is θ=4pin-pi/3 ,θ=4pin+(7pi)/3

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cos(θ/2+pi/2)= 1/2

Solution

cos (\frac{θ}{2} + \frac{π}{2}) = \frac{1}{2}

Solution

θ = 4 πn - \frac{π}{3}, θ = 4 πn + \frac{7 π}{3}

Degrees

θ = - 6 0^{\circ} + 72 0^{\circ} n, θ = 42 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{θ}{2} + \frac{π}{2}) = \frac{1}{2}

General solutions for

cos (\frac{θ}{2} + \frac{π}{2}) = \frac{1}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{θ}{2} + \frac{π}{2} = \frac{π}{3} + 2 πn, \frac{θ}{2} + \frac{π}{2} = \frac{5 π}{3} + 2 πn

\frac{θ}{2} + \frac{π}{2} = \frac{π}{3} + 2 πn, \frac{θ}{2} + \frac{π}{2} = \frac{5 π}{3} + 2 πn

Solve

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

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= \frac{θ}{2}

Simplify

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

\frac{π}{3} + 2 πn - \frac{π}{2}

Group like terms

= 2 πn + \frac{π}{3} - \frac{π}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= \frac{π 2}{6} - \frac{π 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π 3}{6}

Add similar elements:

2 π - 3 π = - π

= \frac{- π}{6}

Apply the fraction rule:

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

= 2 πn - \frac{π}{6}

\frac{θ}{2} = 2 πn - \frac{π}{6}

\frac{θ}{2} = 2 πn - \frac{π}{6}

\frac{θ}{2} = 2 πn - \frac{π}{6}

Multiply both sides by

2

\frac{θ}{2} = 2 πn - \frac{π}{6}

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn - 2 \cdot \frac{π}{6}

Simplify

\frac{2 θ}{2} = 2 \cdot 2 πn - 2 \cdot \frac{π}{6}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot 2 πn - 2 \cdot \frac{π}{6} : 4 πn - \frac{π}{3}

2 \cdot 2 πn - 2 \cdot \frac{π}{6}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

= 4 πn - \frac{π}{3}

θ = 4 πn - \frac{π}{3}

θ = 4 πn - \frac{π}{3}

θ = 4 πn - \frac{π}{3}

Solve

\frac{θ}{2} + \frac{π}{2} = \frac{5 π}{3} + 2 πn : θ = 4 πn + \frac{7 π}{3}

\frac{θ}{2} + \frac{π}{2} = \frac{5 π}{3} + 2 πn

Move

\frac{π}{2}

to the right side

\frac{θ}{2} + \frac{π}{2} = \frac{5 π}{3} + 2 πn

Subtract

\frac{π}{2}

from both sides

\frac{θ}{2} + \frac{π}{2} - \frac{π}{2} = \frac{5 π}{3} + 2 πn - \frac{π}{2}

Simplify

\frac{θ}{2} + \frac{π}{2} - \frac{π}{2} = \frac{5 π}{3} + 2 πn - \frac{π}{2}

Simplify

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

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

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= \frac{θ}{2}

Simplify

\frac{5 π}{3} + 2 πn - \frac{π}{2} : 2 πn + \frac{7 π}{6}

\frac{5 π}{3} + 2 πn - \frac{π}{2}

Group like terms

= 2 πn - \frac{π}{2} + \frac{5 π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 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

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{5 π}{3} :

multiply the denominator and numerator by

2

\frac{5 π}{3} = \frac{5 π 2}{3 \cdot 2} = \frac{10 π}{6}

= - \frac{π 3}{6} + \frac{10 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 10 π}{6}

Add similar elements:

- 3 π + 10 π = 7 π

= 2 πn + \frac{7 π}{6}

\frac{θ}{2} = 2 πn + \frac{7 π}{6}

\frac{θ}{2} = 2 πn + \frac{7 π}{6}

\frac{θ}{2} = 2 πn + \frac{7 π}{6}

Multiply both sides by

2

\frac{θ}{2} = 2 πn + \frac{7 π}{6}

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{7 π}{6}

Simplify

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{7 π}{6}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot 2 πn + 2 \cdot \frac{7 π}{6} : 4 πn + \frac{7 π}{3}

2 \cdot 2 πn + 2 \cdot \frac{7 π}{6}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

= 4 πn + \frac{7 π}{3}

θ = 4 πn + \frac{7 π}{3}

θ = 4 πn + \frac{7 π}{3}

θ = 4 πn + \frac{7 π}{3}

θ = 4 πn - \frac{π}{3}, θ = 4 πn + \frac{7 π}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ/2+pi/2)= 1/2 ?
The general solution for cos(θ/2+pi/2)= 1/2 is θ=4pin-pi/3 ,θ=4pin+(7pi)/3