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Popular Trigonometry >

2cos(x^2)-sqrt(2)=0

Solution

2 cos (x^{2}) - 2 = 0

Solution

x = \frac{π + 8 πn}{2}, x = - \frac{π + 8 πn}{2}, x = \frac{7 π + 8 πn}{2}, x = - \frac{7 π + 8 πn}{2}

Degrees

x = 0^{\circ} + 152.33118 \dots^{\circ} n, x = 0^{\circ} - 152.33118 \dots^{\circ} n, x = 0^{\circ} + 196.65871 \dots^{\circ} n, x = 0^{\circ} - 196.65871 \dots^{\circ} n

Solution steps

2 cos (x^{2}) - 2 = 0

Move

2

to the right side

2 cos (x^{2}) - 2 = 0

Add

2

to both sides

2 cos (x^{2}) - 2 + 2 = 0 + 2

Simplify

2 cos (x^{2}) = 2

2 cos (x^{2}) = 2

Divide both sides by

2

2 cos (x^{2}) = 2

Divide both sides by

2

\frac{2 cos ( x ^{2} )}{2} = \frac{2}{2}

Simplify

cos (x^{2}) = \frac{2}{2}

cos (x^{2}) = \frac{2}{2}

General solutions for

cos (x^{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}

x^{2} = \frac{π}{4} + 2 πn, x^{2} = \frac{7 π}{4} + 2 πn

x^{2} = \frac{π}{4} + 2 πn, x^{2} = \frac{7 π}{4} + 2 πn

Solve

x^{2} = \frac{π}{4} + 2 πn : x = \frac{π + 8 πn}{2}, x = - \frac{π + 8 πn}{2}

x^{2} = \frac{π}{4} + 2 πn

For

x^{2} = f (a)

the solutions are

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

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

Simplify

\frac{π}{4} + 2 πn : \frac{π + 8 πn}{2}

\frac{π}{4} + 2 πn

Join

\frac{π}{4} + 2 πn : \frac{π + 8 πn}{4}

\frac{π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

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

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 8 πn}{4}

= \frac{π + 8 πn}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{π + 8 πn}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{π + 8 πn}{2}

Simplify

- \frac{π}{4} + 2 πn : - \frac{π + 8 πn}{2}

- \frac{π}{4} + 2 πn

Join

\frac{π}{4} + 2 πn : \frac{π + 8 πn}{4}

\frac{π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

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

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 8 πn}{4}

= - \frac{π + 8 πn}{4}

Simplify

\frac{π + 8 πn}{4} : \frac{π + 8 πn}{2}

\frac{π + 8 πn}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{π + 8 πn}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{π + 8 πn}{2}

= - \frac{π + 8 πn}{2}

x = \frac{π + 8 πn}{2}, x = - \frac{π + 8 πn}{2}

Solve

x^{2} = \frac{7 π}{4} + 2 πn : x = \frac{7 π + 8 πn}{2}, x = - \frac{7 π + 8 πn}{2}

x^{2} = \frac{7 π}{4} + 2 πn

For

x^{2} = f (a)

the solutions are

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

x = \frac{7 π}{4} + 2 πn, x = - \frac{7 π}{4} + 2 πn

Simplify

\frac{7 π}{4} + 2 πn : \frac{7 π + 8 πn}{2}

\frac{7 π}{4} + 2 πn

Join

\frac{7 π}{4} + 2 πn : \frac{7 π + 8 πn}{4}

\frac{7 π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{7 π}{4} + \frac{2 πn \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{7 π + 8 πn}{4}

= \frac{7 π + 8 πn}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{7 π + 8 πn}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{7 π + 8 πn}{2}

Simplify

- \frac{7 π}{4} + 2 πn : - \frac{7 π + 8 πn}{2}

- \frac{7 π}{4} + 2 πn

Join

\frac{7 π}{4} + 2 πn : \frac{7 π + 8 πn}{4}

\frac{7 π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{7 π}{4} + \frac{2 πn \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{7 π + 8 πn}{4}

= - \frac{7 π + 8 πn}{4}

Simplify

\frac{7 π + 8 πn}{4} : \frac{7 π + 8 πn}{2}

\frac{7 π + 8 πn}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{7 π + 8 πn}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{7 π + 8 πn}{2}

= - \frac{7 π + 8 πn}{2}

x = \frac{7 π + 8 πn}{2}, x = - \frac{7 π + 8 πn}{2}

x = \frac{π + 8 πn}{2}, x = - \frac{π + 8 πn}{2}, x = \frac{7 π + 8 πn}{2}, x = - \frac{7 π + 8 πn}{2}

Graph

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