What is the general solution for cos(θ/2)=-(sqrt(2))/2 ?

The general solution for cos(θ/2)=-(sqrt(2))/2 is θ=(3pi)/2+4pin,θ=(5pi)/2+4pin

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cos(θ/2)=-(sqrt(2))/2

Solution

cos (\frac{θ}{2}) = - \frac{2}{2}

Solution

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

Degrees

θ = 27 0^{\circ} + 72 0^{\circ} n, θ = 45 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{θ}{2}) = - \frac{2}{2}

General solutions for

cos (\frac{θ}{2}) = - \frac{2}{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{3 π}{4} + 2 πn, \frac{θ}{2} = \frac{5 π}{4} + 2 πn

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

Solve

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

\frac{θ}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{3 π}{2} + 4 πn

θ = \frac{3 π}{2} + 4 πn

θ = \frac{3 π}{2} + 4 πn

θ = \frac{3 π}{2} + 4 πn

Solve

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

\frac{θ}{2} = \frac{5 π}{4} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{5 π}{4} + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn : \frac{5 π}{2} + 4 πn

2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{4} = \frac{5 π}{2}

2 \cdot \frac{5 π}{4}

Multiply fractions:

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

= \frac{5 π 2}{4}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{4}

Cancel the common factor:

2

= \frac{5 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{2} + 4 πn

θ = \frac{5 π}{2} + 4 πn

θ = \frac{5 π}{2} + 4 πn

θ = \frac{5 π}{2} + 4 πn

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

Popular Examples

16sin^2(x)+24sin(x)+8=0

16 sin^{2} (x) + 24 sin (x) + 8 = 0

1-3tan^2(x)=0

1 - 3 tan^{2} (x) = 0

6tan^2(θ)-10tan(θ)+1=-5tan(θ)

6 tan^{2} (θ) - 10 tan (θ) + 1 = - 5 tan (θ)

2sin(2x)+6sin(x)-2cos(x)=3

2 sin (2 x) + 6 sin (x) - 2 cos (x) = 3

sin(2x)= 4/5

sin (2 x) = \frac{4}{5}

Frequently Asked Questions (FAQ)

What is the general solution for cos(θ/2)=-(sqrt(2))/2 ?
The general solution for cos(θ/2)=-(sqrt(2))/2 is θ=(3pi)/2+4pin,θ=(5pi)/2+4pin