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Popular >

6cos^2(2x)+10cos(4x)=12.05718

Solution

6 cos^{2} (2 x) + 10 cos (4 x) = 12.05718

Solution

x = \frac{0.40000 \dots}{2} + πn, x = π - \frac{0.40000 \dots}{2} + πn, x = \frac{2.74159 \dots}{2} + πn, x = - \frac{2.74159 \dots}{2} + πn

Degrees

x = 11.45916 \dots^{\circ} + 18 0^{\circ} n, x = 168.54083 \dots^{\circ} + 18 0^{\circ} n, x = 78.54083 \dots^{\circ} + 18 0^{\circ} n, x = - 78.54083 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 cos^{2} (2 x) + 10 cos (4 x) = 12.05718

Subtract

12.05718

from both sides

6 cos^{2} (2 x) + 10 cos (4 x) - 12.05718 = 0

Rewrite using trig identities

- 12.05718 + 10 cos (4 x) + 6 cos^{2} (2 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

= - 12.05718 + 10 (2 cos^{2} (2 x) - 1) + 6 cos^{2} (2 x)

Simplify

- 12.05718 + 10 (2 cos^{2} (2 x) - 1) + 6 cos^{2} (2 x) : 26 cos^{2} (2 x) - 22.05718

- 12.05718 + 10 (2 cos^{2} (2 x) - 1) + 6 cos^{2} (2 x)

Expand

10 (2 cos^{2} (2 x) - 1) : 20 cos^{2} (2 x) - 10

10 (2 cos^{2} (2 x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 10, b = 2 cos^{2} (2 x), c = 1

= 10 \cdot 2 cos^{2} (2 x) - 10 \cdot 1

Simplify

10 \cdot 2 cos^{2} (2 x) - 10 \cdot 1 : 20 cos^{2} (2 x) - 10

10 \cdot 2 cos^{2} (2 x) - 10 \cdot 1

Multiply the numbers:

10 \cdot 2 = 20

= 20 cos^{2} (2 x) - 10 \cdot 1

Multiply the numbers:

10 \cdot 1 = 10

= 20 cos^{2} (2 x) - 10

= 20 cos^{2} (2 x) - 10

= - 12.05718 + 20 cos^{2} (2 x) - 10 + 6 cos^{2} (2 x)

Simplify

- 12.05718 + 20 cos^{2} (2 x) - 10 + 6 cos^{2} (2 x) : 26 cos^{2} (2 x) - 22.05718

- 12.05718 + 20 cos^{2} (2 x) - 10 + 6 cos^{2} (2 x)

Group like terms

= 20 cos^{2} (2 x) + 6 cos^{2} (2 x) - 12.05718 - 10

Add similar elements:

20 cos^{2} (2 x) + 6 cos^{2} (2 x) = 26 cos^{2} (2 x)

= 26 cos^{2} (2 x) - 12.05718 - 10

Subtract the numbers:

- 12.05718 - 10 = - 22.05718

= 26 cos^{2} (2 x) - 22.05718

= 26 cos^{2} (2 x) - 22.05718

= 26 cos^{2} (2 x) - 22.05718

- 22.05718 + 26 cos^{2} (2 x) = 0

Solve by substitution

- 22.05718 + 26 cos^{2} (2 x) = 0

Let:

cos (2 x) = u

- 22.05718 + 26 u^{2} = 0

- 22.05718 + 26 u^{2} = 0 : u = \frac{22.05718}{26}, u = - \frac{22.05718}{26}

- 22.05718 + 26 u^{2} = 0

Move

22.05718

to the right side

- 22.05718 + 26 u^{2} = 0

Add

22.05718

to both sides

- 22.05718 + 26 u^{2} + 22.05718 = 0 + 22.05718

Simplify

26 u^{2} = 22.05718

26 u^{2} = 22.05718

Divide both sides by

26

26 u^{2} = 22.05718

Divide both sides by

26

\frac{26 u ^{2}}{26} = \frac{22.05718}{26}

Simplify

u^{2} = \frac{22.05718}{26}

u^{2} = \frac{22.05718}{26}

For

x^{2} = f (a)

the solutions are

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

u = \frac{22.05718}{26}, u = - \frac{22.05718}{26}

Substitute back

u = cos (2 x)

cos (2 x) = \frac{22.05718}{26}, cos (2 x) = - \frac{22.05718}{26}

cos (2 x) = \frac{22.05718}{26}, cos (2 x) = - \frac{22.05718}{26}

cos (2 x) = \frac{22.05718}{26} : x = \frac{arccos ( \frac{22.05718}{26} )}{2} + πn, x = π - \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

cos (2 x) = \frac{22.05718}{26}

Apply trig inverse properties

cos (2 x) = \frac{22.05718}{26}

General solutions for

cos (2 x) = \frac{22.05718}{26}

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

2 x = arccos (\frac{22.05718}{26}) + 2 πn, 2 x = 2 π - arccos (\frac{22.05718}{26}) + 2 πn

2 x = arccos (\frac{22.05718}{26}) + 2 πn, 2 x = 2 π - arccos (\frac{22.05718}{26}) + 2 πn

Solve

2 x = arccos (\frac{22.05718}{26}) + 2 πn : x = \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

2 x = arccos (\frac{22.05718}{26}) + 2 πn

Divide both sides by

2

2 x = arccos (\frac{22.05718}{26}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{22.05718}{26} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

x = \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

Solve

2 x = 2 π - arccos (\frac{22.05718}{26}) + 2 πn : x = π - \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

2 x = 2 π - arccos (\frac{22.05718}{26}) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (\frac{22.05718}{26}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{22.05718}{26} )}{2} + \frac{2 πn}{2}

Simplify

x = π - \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

x = π - \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

x = \frac{arccos ( \frac{22.05718}{26} )}{2} + πn, x = π - \frac{arccos ( \frac{22.05718}{26} )}{2} + πn

cos (2 x) = - \frac{22.05718}{26} : x = \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn, x = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

cos (2 x) = - \frac{22.05718}{26}

Apply trig inverse properties

cos (2 x) = - \frac{22.05718}{26}

General solutions for

cos (2 x) = - \frac{22.05718}{26}

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

2 x = arccos (- \frac{22.05718}{26}) + 2 πn, 2 x = - arccos (- \frac{22.05718}{26}) + 2 πn

2 x = arccos (- \frac{22.05718}{26}) + 2 πn, 2 x = - arccos (- \frac{22.05718}{26}) + 2 πn

Solve

2 x = arccos (- \frac{22.05718}{26}) + 2 πn : x = \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

2 x = arccos (- \frac{22.05718}{26}) + 2 πn

Divide both sides by

2

2 x = arccos (- \frac{22.05718}{26}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( - \frac{22.05718}{26} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

x = \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

Solve

2 x = - arccos (- \frac{22.05718}{26}) + 2 πn : x = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

2 x = - arccos (- \frac{22.05718}{26}) + 2 πn

Divide both sides by

2

2 x = - arccos (- \frac{22.05718}{26}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

x = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

x = \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn, x = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

Combine all the solutions

x = \frac{arccos ( \frac{22.05718}{26} )}{2} + πn, x = π - \frac{arccos ( \frac{22.05718}{26} )}{2} + πn, x = \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn, x = - \frac{arccos ( - \frac{22.05718}{26} )}{2} + πn

Show solutions in decimal form

x = \frac{0.40000 \dots}{2} + πn, x = π - \frac{0.40000 \dots}{2} + πn, x = \frac{2.74159 \dots}{2} + πn, x = - \frac{2.74159 \dots}{2} + πn

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