What is the general solution for 8*cos^2(3x)+6*cos(6x)=10216 ?

The general solution for 8*cos^2(3x)+6*cos(6x)=10216 is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

8cos^2(3x)+6cos(6x)=10216

Solution

8 \cdot cos^{2} (3 x) + 6 \cdot cos (6 x) = 10216

Solution

No Solution for x \in R

Solution steps

8 cos^{2} (3 x) + 6 cos (6 x) = 10216

Subtract

10216

from both sides

8 cos^{2} (3 x) + 6 cos (6 x) - 10216 = 0

Rewrite using trig identities

- 10216 + 6 cos (6 x) + 8 cos^{2} (3 x)

Use the following identity:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

= - 10216 + 6 cos (6 x) + 8 \cdot \frac{1 + cos ( 2 \cdot 3 x )}{2}

Simplify

- 10216 + 6 cos (6 x) + 8 \cdot \frac{1 + cos ( 2 \cdot 3 x )}{2} : 10 cos (6 x) - 10212

- 10216 + 6 cos (6 x) + 8 \cdot \frac{1 + cos ( 2 \cdot 3 x )}{2}

8 \cdot \frac{1 + cos ( 2 \cdot 3 x )}{2} = 4 (cos (6 x) + 1)

8 \cdot \frac{1 + cos ( 2 \cdot 3 x )}{2}

Multiply the numbers:

2 \cdot 3 = 6

= 8 \cdot \frac{cos ( 6 x ) + 1}{2}

Multiply fractions:

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

= \frac{( 1 + cos ( 6 x ) ) \cdot 8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 (cos (6 x) + 1)

= - 10216 + 6 cos (6 x) + 4 (cos (6 x) + 1)

Expand

4 (cos (6 x) + 1) : 4 cos (6 x) + 4

4 (cos (6 x) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 4, b = cos (6 x), c = 1

= 4 cos (6 x) + 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 4 cos (6 x) + 4

= - 10216 + 6 cos (6 x) + 4 cos (6 x) + 4

Simplify

- 10216 + 6 cos (6 x) + 4 cos (6 x) + 4 : 10 cos (6 x) - 10212

- 10216 + 6 cos (6 x) + 4 cos (6 x) + 4

Add similar elements:

6 cos (6 x) + 4 cos (6 x) = 10 cos (6 x)

= - 10216 + 10 cos (6 x) + 4

Group like terms

= 10 cos (6 x) - 10216 + 4

Add/Subtract the numbers:

- 10216 + 4 = - 10212

= 10 cos (6 x) - 10212

= 10 cos (6 x) - 10212

= 10 cos (6 x) - 10212

- 10212 + 10 cos (6 x) = 0

Move

10212

to the right side

- 10212 + 10 cos (6 x) = 0

Add

10212

to both sides

- 10212 + 10 cos (6 x) + 10212 = 0 + 10212

Simplify

10 cos (6 x) = 10212

10 cos (6 x) = 10212

Divide both sides by

10

10 cos (6 x) = 10212

Divide both sides by

10

\frac{10 cos ( 6 x )}{10} = \frac{10212}{10}

Simplify

cos (6 x) = \frac{5106}{5}

cos (6 x) = \frac{5106}{5}

- 1 \leq cos (x) \leq 1

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 8*cos^2(3x)+6*cos(6x)=10216 ?
The general solution for 8*cos^2(3x)+6*cos(6x)=10216 is No Solution for x\in\mathbb{R}