What is the general solution for cos^2(x/3)-sin^2(x/3)=-1/2 ?

The general solution for cos^2(x/3)-sin^2(x/3)=-1/2 is x=pi+3pin,x=2pi+3pin

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cos^2(x/3)-sin^2(x/3)=-1/2

Solution

cos^{2} (\frac{x}{3}) - sin^{2} (\frac{x}{3}) = - \frac{1}{2}

Solution

x = π + 3 πn, x = 2 π + 3 πn

Degrees

x = 18 0^{\circ} + 54 0^{\circ} n, x = 36 0^{\circ} + 54 0^{\circ} n

Solution steps

cos^{2} (\frac{x}{3}) - sin^{2} (\frac{x}{3}) = - \frac{1}{2}

Rewrite using trig identities

cos^{2} (\frac{x}{3}) - sin^{2} (\frac{x}{3})

Use the Double Angle identity:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

= cos (2 \cdot \frac{x}{3})

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

General solutions for

cos (2 \cdot \frac{x}{3}) = - \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}

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

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

Solve

2 \cdot \frac{x}{3} = \frac{2 π}{3} + 2 πn : x = π + 3 πn

2 \cdot \frac{x}{3} = \frac{2 π}{3} + 2 πn

Multiply fractions:

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

\frac{2 x}{3} = \frac{2 π}{3} + 2 πn

Multiply both sides by

3

\frac{2 x}{3} = \frac{2 π}{3} + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 x}{3} = 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 x}{3} = 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 x

Simplify

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

3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 3}{3}

Cancel the common factor:

3

= 2 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 2 π + 6 πn

2 x = 2 π + 6 πn

2 x = 2 π + 6 πn

2 x = 2 π + 6 πn

Divide both sides by

2

2 x = 2 π + 6 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} + \frac{6 πn}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} + \frac{6 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{2 π}{2} + \frac{6 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= π + \frac{6 πn}{2}

Divide the numbers:

\frac{6}{2} = 3

= π + 3 πn

x = π + 3 πn

x = π + 3 πn

x = π + 3 πn

Solve

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

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

Multiply fractions:

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

\frac{2 x}{3} = \frac{4 π}{3} + 2 πn

Multiply both sides by

3

\frac{2 x}{3} = \frac{4 π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 x

Simplify

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

3 \cdot \frac{4 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{4 π}{3} = 4 π

3 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 3}{3}

Cancel the common factor:

3

= 4 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 4 π + 6 πn

2 x = 4 π + 6 πn

2 x = 4 π + 6 πn

2 x = 4 π + 6 πn

Divide both sides by

2

2 x = 4 π + 6 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{4 π}{2} + \frac{6 πn}{2}

Simplify

\frac{2 x}{2} = \frac{4 π}{2} + \frac{6 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{4 π}{2} + \frac{6 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π + \frac{6 πn}{2}

Divide the numbers:

\frac{6}{2} = 3

= 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

x = π + 3 πn, x = 2 π + 3 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x/3)-sin^2(x/3)=-1/2 ?
The general solution for cos^2(x/3)-sin^2(x/3)=-1/2 is x=pi+3pin,x=2pi+3pin