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Popular >

2cos^2(2x)+9cos(4x)=-2.4214900000000…

Solution

2 \cdot cos^{2} (2 x) + 9 \cdot cos (4 x) = - 2.4214900000000 \dots

Solution

x = \frac{0.95999 \dots}{2} + πn, x = π - \frac{0.95999 \dots}{2} + πn, x = \frac{2.18159 \dots}{2} + πn, x = - \frac{2.18159 \dots}{2} + πn

Degrees

x = 27.50196 \dots^{\circ} + 18 0^{\circ} n, x = 152.49803 \dots^{\circ} + 18 0^{\circ} n, x = 62.49803 \dots^{\circ} + 18 0^{\circ} n, x = - 62.49803 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 cos^{2} (2 x) + 9 cos (4 x) = - 2.42149 \dots

Subtract

- 2.42149 \dots

from both sides

2 cos^{2} (2 x) + 9 cos (4 x) + 2.42149 \dots = 0

Rewrite using trig identities

2.42149 \dots + 2 cos^{2} (2 x) + 9 cos (4 x)

cos (4 x) = 2 cos^{2} (2 x) - 1

cos (4 x)

Rewrite as

= cos (2 \cdot 2 x)

Use the Double Angle identity:

cos (2 x) = 2 cos^{2} (x) - 1

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

= 2 cos^{2} (2 x) - 1

= 2.42149 \dots + 2 cos^{2} (2 x) + 9 (2 cos^{2} (2 x) - 1)

Simplify

2.42149 \dots + 2 cos^{2} (2 x) + 9 (2 cos^{2} (2 x) - 1) : 20 cos^{2} (2 x) - 6.57851

2.42149 \dots + 2 cos^{2} (2 x) + 9 (2 cos^{2} (2 x) - 1)

Expand

9 (2 cos^{2} (2 x) - 1) : 18 cos^{2} (2 x) - 9

9 (2 cos^{2} (2 x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 2 cos^{2} (2 x), c = 1

= 9 \cdot 2 cos^{2} (2 x) - 9 \cdot 1

Simplify

9 \cdot 2 cos^{2} (2 x) - 9 \cdot 1 : 18 cos^{2} (2 x) - 9

9 \cdot 2 cos^{2} (2 x) - 9 \cdot 1

Multiply the numbers:

9 \cdot 2 = 18

= 18 cos^{2} (2 x) - 9 \cdot 1

Multiply the numbers:

9 \cdot 1 = 9

= 18 cos^{2} (2 x) - 9

= 18 cos^{2} (2 x) - 9

= 2.42149 \dots + 2 cos^{2} (2 x) + 18 cos^{2} (2 x) - 9

Simplify

2.42149 \dots + 2 cos^{2} (2 x) + 18 cos^{2} (2 x) - 9 : 20 cos^{2} (2 x) - 6.57851

2.42149 \dots + 2 cos^{2} (2 x) + 18 cos^{2} (2 x) - 9

Add similar elements:

2 cos^{2} (2 x) + 18 cos^{2} (2 x) = 20 cos^{2} (2 x)

= 2.42149 \dots + 20 cos^{2} (2 x) - 9

Group like terms

= 20 cos^{2} (2 x) + 2.42149 \dots - 9

Add/Subtract the numbers:

2.42149 \dots - 9 = - 6.57851

= 20 cos^{2} (2 x) - 6.57851

= 20 cos^{2} (2 x) - 6.57851

= 20 cos^{2} (2 x) - 6.57851

- 6.57851 + 20 cos^{2} (2 x) = 0

Solve by substitution

- 6.57851 + 20 cos^{2} (2 x) = 0

Let:

cos (2 x) = u

- 6.57851 + 20 u^{2} = 0

- 6.57851 + 20 u^{2} = 0 : u = \frac{32.89255}{10}, u = - \frac{32.89255}{10}

- 6.57851 + 20 u^{2} = 0

Move

6.57851

to the right side

- 6.57851 + 20 u^{2} = 0

Add

6.57851

to both sides

- 6.57851 + 20 u^{2} + 6.57851 = 0 + 6.57851

Simplify

20 u^{2} = 6.57851

20 u^{2} = 6.57851

Divide both sides by

20

20 u^{2} = 6.57851

Divide both sides by

20

\frac{20 u ^{2}}{20} = \frac{6.57851}{20}

Simplify

u^{2} = \frac{6.57851}{20}

u^{2} = \frac{6.57851}{20}

For

x^{2} = f (a)

the solutions are

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

u = \frac{6.57851}{20}, u = - \frac{6.57851}{20}

\frac{6.57851}{20} = \frac{32.89255}{10}

\frac{6.57851}{20}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{6.57851}{20}

20 = 25

20

Prime factorization of

20 : 2^{2} \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

= 5 2^{2}

Apply radical rule:

2^{2} = 2

= 25

= \frac{6.57851}{2 5}

Rationalize

\frac{6.57851}{2 5} : \frac{32.89255}{10}

\frac{6.57851}{2 5}

Multiply by the conjugate

\frac{5}{5}

= \frac{6.57851 5}{2 5 5}

6.57851 5 = 32.89255

6.57851 5

Apply radical rule:

a b = a \cdot b

5 6.57851 = 5 \cdot 6.57851

= 5 \cdot 6.57851

Multiply the numbers:

6.57851 \cdot 5 = 32.89255

= 32.89255

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= \frac{32.89255}{10}

= \frac{32.89255}{10}

- \frac{6.57851}{20} = - \frac{32.89255}{10}

- \frac{6.57851}{20}

Simplify

\frac{6.57851}{20} : \frac{6.57851}{2 5}

\frac{6.57851}{20}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{6.57851}{20}

20 = 25

20

Prime factorization of

20 : 2^{2} \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

= 5 2^{2}

Apply radical rule:

2^{2} = 2

= 25

= \frac{6.57851}{2 5}

= - \frac{6.57851}{2 5}

Rationalize

- \frac{6.57851}{2 5} : - \frac{32.89255}{10}

- \frac{6.57851}{2 5}

Multiply by the conjugate

\frac{5}{5}

= - \frac{6.57851 5}{2 5 5}

6.57851 5 = 32.89255

6.57851 5

Apply radical rule:

a b = a \cdot b

5 6.57851 = 5 \cdot 6.57851

= 5 \cdot 6.57851

Multiply the numbers:

6.57851 \cdot 5 = 32.89255

= 32.89255

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= - \frac{32.89255}{10}

= - \frac{32.89255}{10}

u = \frac{32.89255}{10}, u = - \frac{32.89255}{10}

Substitute back

u = cos (2 x)

cos (2 x) = \frac{32.89255}{10}, cos (2 x) = - \frac{32.89255}{10}

cos (2 x) = \frac{32.89255}{10}, cos (2 x) = - \frac{32.89255}{10}

cos (2 x) = \frac{32.89255}{10} : x = \frac{arccos ( \frac{32.89255}{10} )}{2} + πn, x = π - \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

cos (2 x) = \frac{32.89255}{10}

Apply trig inverse properties

cos (2 x) = \frac{32.89255}{10}

General solutions for

cos (2 x) = \frac{32.89255}{10}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x = arccos (\frac{32.89255}{10}) + 2 πn, 2 x = 2 π - arccos (\frac{32.89255}{10}) + 2 πn

2 x = arccos (\frac{32.89255}{10}) + 2 πn, 2 x = 2 π - arccos (\frac{32.89255}{10}) + 2 πn

Solve

2 x = arccos (\frac{32.89255}{10}) + 2 πn : x = \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

2 x = arccos (\frac{32.89255}{10}) + 2 πn

Divide both sides by

2

2 x = arccos (\frac{32.89255}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{32.89255}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

x = \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

Solve

2 x = 2 π - arccos (\frac{32.89255}{10}) + 2 πn : x = π - \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

2 x = 2 π - arccos (\frac{32.89255}{10}) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (\frac{32.89255}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{32.89255}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = π - \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

x = π - \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

x = \frac{arccos ( \frac{32.89255}{10} )}{2} + πn, x = π - \frac{arccos ( \frac{32.89255}{10} )}{2} + πn

cos (2 x) = - \frac{32.89255}{10} : x = \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn, x = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

cos (2 x) = - \frac{32.89255}{10}

Apply trig inverse properties

cos (2 x) = - \frac{32.89255}{10}

General solutions for

cos (2 x) = - \frac{32.89255}{10}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

2 x = arccos (- \frac{32.89255}{10}) + 2 πn, 2 x = - arccos (- \frac{32.89255}{10}) + 2 πn

2 x = arccos (- \frac{32.89255}{10}) + 2 πn, 2 x = - arccos (- \frac{32.89255}{10}) + 2 πn

Solve

2 x = arccos (- \frac{32.89255}{10}) + 2 πn : x = \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

2 x = arccos (- \frac{32.89255}{10}) + 2 πn

Divide both sides by

2

2 x = arccos (- \frac{32.89255}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( - \frac{32.89255}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

x = \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

Solve

2 x = - arccos (- \frac{32.89255}{10}) + 2 πn : x = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

2 x = - arccos (- \frac{32.89255}{10}) + 2 πn

Divide both sides by

2

2 x = - arccos (- \frac{32.89255}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

x = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

x = \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn, x = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

Combine all the solutions

x = \frac{arccos ( \frac{32.89255}{10} )}{2} + πn, x = π - \frac{arccos ( \frac{32.89255}{10} )}{2} + πn, x = \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn, x = - \frac{arccos ( - \frac{32.89255}{10} )}{2} + πn

Show solutions in decimal form

x = \frac{0.95999 \dots}{2} + πn, x = π - \frac{0.95999 \dots}{2} + πn, x = \frac{2.18159 \dots}{2} + πn, x = - \frac{2.18159 \dots}{2} + πn

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2*cos^2(2x)+9*cos(4x)=-2.4214900000000…

Solution

Solution

Popular Examples

2cos^2(2x)+9cos(4x)=-2.4214900000000…