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solvefor x,cos^2(xy)= 1/4

Solution

solve for

x, cos^{2} (x y) = \frac{1}{4}

Solution

x = \frac{π}{3 y} + \frac{2 πn}{y}, x = \frac{5 π}{3 y} + \frac{2 πn}{y}, x = \frac{2 π}{3 y} + \frac{2 πn}{y}, x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

Solution steps

cos^{2} (x y) = \frac{1}{4}

Solve by substitution

cos^{2} (x y) = \frac{1}{4}

Let:

cos (x y) = u

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = cos (x y)

cos (x y) = \frac{1}{2}, cos (x y) = - \frac{1}{2}

cos (x y) = \frac{1}{2}, cos (x y) = - \frac{1}{2}

cos (x y) = \frac{1}{2} : x = \frac{π}{3 y} + \frac{2 πn}{y}, x = \frac{5 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

cos (x y) = \frac{1}{2}

General solutions for

cos (x y) = \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}

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

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

Solve

x y = \frac{π}{3} + 2 πn : x = \frac{π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x y = \frac{π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

x y = \frac{π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

\frac{x y}{y} = \frac{\frac{π}{3}}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

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

Simplify

\frac{x y}{y} : x

\frac{x y}{y}

Cancel the common factor:

y

= x

Simplify

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

\frac{\frac{π}{3}}{y} + \frac{2 πn}{y}

Apply the fraction rule:

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

= \frac{π}{3 y} + \frac{2 πn}{y}

x = \frac{π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{π}{3 y} + \frac{2 πn}{y}; y \neq = 0

Solve

x y = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x y = \frac{5 π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

x y = \frac{5 π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

\frac{x y}{y} = \frac{\frac{5 π}{3}}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

\frac{x y}{y} = \frac{\frac{5 π}{3}}{y} + \frac{2 πn}{y}

Simplify

\frac{x y}{y} : x

\frac{x y}{y}

Cancel the common factor:

y

= x

Simplify

\frac{\frac{5 π}{3}}{y} + \frac{2 πn}{y} : \frac{5 π}{3 y} + \frac{2 πn}{y}

\frac{\frac{5 π}{3}}{y} + \frac{2 πn}{y}

Apply the fraction rule:

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

= \frac{5 π}{3 y} + \frac{2 πn}{y}

x = \frac{5 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{5 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{5 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{π}{3 y} + \frac{2 πn}{y}, x = \frac{5 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

cos (x y) = - \frac{1}{2} : x = \frac{2 π}{3 y} + \frac{2 πn}{y}, x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

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

General solutions for

cos (x y) = - \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}

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

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

Solve

x y = \frac{2 π}{3} + 2 πn : x = \frac{2 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x y = \frac{2 π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

x y = \frac{2 π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

\frac{x y}{y} = \frac{\frac{2 π}{3}}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

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

Simplify

\frac{x y}{y} : x

\frac{x y}{y}

Cancel the common factor:

y

= x

Simplify

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

\frac{\frac{2 π}{3}}{y} + \frac{2 πn}{y}

Apply the fraction rule:

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

= \frac{2 π}{3 y} + \frac{2 πn}{y}

x = \frac{2 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{2 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{2 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

Solve

x y = \frac{4 π}{3} + 2 πn : x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x y = \frac{4 π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

x y = \frac{4 π}{3} + 2 πn

Divide both sides by

y; y \neq = 0

\frac{x y}{y} = \frac{\frac{4 π}{3}}{y} + \frac{2 πn}{y}; y \neq = 0

Simplify

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

Simplify

\frac{x y}{y} : x

\frac{x y}{y}

Cancel the common factor:

y

= x

Simplify

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

\frac{\frac{4 π}{3}}{y} + \frac{2 πn}{y}

Apply the fraction rule:

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

= \frac{4 π}{3 y} + \frac{2 πn}{y}

x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

x = \frac{2 π}{3 y} + \frac{2 πn}{y}, x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

Combine all the solutions

x = \frac{π}{3 y} + \frac{2 πn}{y}, x = \frac{5 π}{3 y} + \frac{2 πn}{y}, x = \frac{2 π}{3 y} + \frac{2 πn}{y}, x = \frac{4 π}{3 y} + \frac{2 πn}{y}; y \neq = 0

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