What is the general solution for cos(2x)+7cos(x)+4=0 ?

The general solution for cos(2x)+7cos(x)+4=0 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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cos(2x)+7cos(x)+4=0

Solution

cos (2 x) + 7 cos (x) + 4 = 0

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (2 x) + 7 cos (x) + 4 = 0

Rewrite using trig identities

4 + cos (2 x) + 7 cos (x)

Use the Double Angle identity:

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

= 4 + 2 cos^{2} (x) - 1 + 7 cos (x)

Simplify

4 + 2 cos^{2} (x) - 1 + 7 cos (x) : 2 cos^{2} (x) + 7 cos (x) + 3

4 + 2 cos^{2} (x) - 1 + 7 cos (x)

Group like terms

= 2 cos^{2} (x) + 7 cos (x) + 4 - 1

Add/Subtract the numbers:

4 - 1 = 3

= 2 cos^{2} (x) + 7 cos (x) + 3

= 2 cos^{2} (x) + 7 cos (x) + 3

3 + 2 cos^{2} (x) + 7 cos (x) = 0

Solve by substitution

3 + 2 cos^{2} (x) + 7 cos (x) = 0

Let:

cos (x) = u

3 + 2 u^{2} + 7 u = 0

3 + 2 u^{2} + 7 u = 0 : u = - \frac{1}{2}, u = - 3

3 + 2 u^{2} + 7 u = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} + 7 u + 3 = 0

Solve with the quadratic formula

2 u^{2} + 7 u + 3 = 0

Quadratic Equation Formula:

For

a = 2, b = 7, c = 3

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

7^{2} - 4 \cdot 2 \cdot 3 = 5

7^{2} - 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 7 \pm 5}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 7 + 5}{2 \cdot 2}, u_{2} = \frac{- 7 - 5}{2 \cdot 2}

u = \frac{- 7 + 5}{2 \cdot 2} : - \frac{1}{2}

\frac{- 7 + 5}{2 \cdot 2}

Add/Subtract the numbers:

- 7 + 5 = - 2

= \frac{- 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = \frac{- 7 - 5}{2 \cdot 2} : - 3

\frac{- 7 - 5}{2 \cdot 2}

Subtract the numbers:

- 7 - 5 = - 12

= \frac{- 12}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 12}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= - 3

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = - 3

Substitute back

u = cos (x)

cos (x) = - \frac{1}{2}, cos (x) = - 3

cos (x) = - \frac{1}{2}, cos (x) = - 3

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

General solutions for

cos (x) = - \frac{1}{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 = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - 3 :

No Solution

cos (x) = - 3

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)+7cos(x)+4=0 ?
The general solution for cos(2x)+7cos(x)+4=0 is x=(2pi)/3+2pin,x=(4pi)/3+2pin