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Popular >

cos^4(2x)= 9/16

Solution

cos^{4} (2 x) = \frac{9}{16}

Solution

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn, x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

Degrees

x = 1 5^{\circ} + 18 0^{\circ} n, x = 16 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n, x = 10 5^{\circ} + 18 0^{\circ} n

Solution steps

cos^{4} (2 x) = \frac{9}{16}

Solve by substitution

cos^{4} (2 x) = \frac{9}{16}

Let:

cos (2 x) = u

u^{4} = \frac{9}{16}

u^{4} = \frac{9}{16} : u = \frac{3}{2}, u = - \frac{3}{2}, u = i \frac{3}{2}, u = - i \frac{3}{2}

u^{4} = \frac{9}{16}

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} = \frac{9}{16}

Solve

v^{2} = \frac{9}{16} : v = \frac{9}{16}, v = - \frac{9}{16}

v^{2} = \frac{9}{16}

For

(g (x))^{2} = f (a)

the solutions are

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

v = \frac{9}{16}, v = - \frac{9}{16}

v = \frac{9}{16}, v = - \frac{9}{16}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{9}{16} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{9}{16}

Simplify

\frac{9}{16} : \frac{3}{4}

\frac{9}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{9}{4}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{4}

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

Solve

u^{2} = - \frac{9}{16} : u = i \frac{3}{2}, u = - i \frac{3}{2}

u^{2} = - \frac{9}{16}

Simplify

\frac{9}{16} : \frac{3}{4}

\frac{9}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{9}{4}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{4}

u^{2} = - \frac{3}{4}

For

x^{2} = f (a)

the solutions are

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

u = - \frac{3}{4}, u = - - \frac{3}{4}

Simplify

- \frac{3}{4} : i \frac{3}{2}

- \frac{3}{4}

Apply radical rule:

- a = - 1 a

- \frac{3}{4} = - 1 \frac{3}{4}

= - 1 \frac{3}{4}

Apply imaginary number rule:

- 1 = i

= i \frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{3}{4} = \frac{3}{4}

= i \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= i \frac{3}{2}

Rewrite

i \frac{3}{2}

in standard complex form:

\frac{3}{2} i

i \frac{3}{2}

Multiply fractions:

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

= \frac{3 i}{2}

= \frac{3}{2} i

Simplify

- - \frac{3}{4} : - i \frac{3}{2}

- - \frac{3}{4}

Simplify

- \frac{3}{4} : i \frac{3}{2}

- \frac{3}{4}

Apply radical rule:

- a = - 1 a

- \frac{3}{4} = - 1 \frac{3}{4}

= - 1 \frac{3}{4}

Apply imaginary number rule:

- 1 = i

= i \frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{3}{4} = \frac{3}{4}

= i \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= i \frac{3}{2}

= - \frac{3}{2} i

= - i \frac{3}{2}

u = i \frac{3}{2}, u = - i \frac{3}{2}

The solutions are

u = \frac{3}{2}, u = - \frac{3}{2}, u = i \frac{3}{2}, u = - i \frac{3}{2}

Substitute back

u = cos (2 x)

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

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

cos (2 x) = \frac{3}{2} : x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn

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

General solutions for

cos (2 x) = \frac{3}{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 x = \frac{π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

Solve

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

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

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

Solve

2 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{12} + πn

2 x = \frac{11 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{11 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2} : \frac{11 π}{12} + πn

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{11 π}{6}}{2} = \frac{11 π}{12}

\frac{\frac{11 π}{6}}{2}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{11 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn

cos (2 x) = - \frac{3}{2} : x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

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

General solutions for

cos (2 x) = - \frac{3}{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 x = \frac{5 π}{6} + 2 πn, 2 x = \frac{7 π}{6} + 2 πn

2 x = \frac{5 π}{6} + 2 πn, 2 x = \frac{7 π}{6} + 2 πn

Solve

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

Solve

2 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{12} + πn

2 x = \frac{7 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2} : \frac{7 π}{12} + πn

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{6}}{2} = \frac{7 π}{12}

\frac{\frac{7 π}{6}}{2}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{7 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

cos (2 x) = i \frac{3}{2} :

No Solution

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

No Solution

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

No Solution

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

No Solution

Combine all the solutions

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn, x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

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