What is the general solution for cos(2x)+5cos(x)+3=0 ?

The general solution for cos(2x)+5cos(x)+3=0 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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cos(2x)+5cos(x)+3=0

Solution

cos (2 x) + 5 cos (x) + 3 = 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) + 5 cos (x) + 3 = 0

Rewrite using trig identities

3 + cos (2 x) + 5 cos (x)

Use the Double Angle identity:

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

= 3 + 2 cos^{2} (x) - 1 + 5 cos (x)

Simplify

3 + 2 cos^{2} (x) - 1 + 5 cos (x) : 2 cos^{2} (x) + 5 cos (x) + 2

3 + 2 cos^{2} (x) - 1 + 5 cos (x)

Group like terms

= 2 cos^{2} (x) + 5 cos (x) + 3 - 1

Add/Subtract the numbers:

3 - 1 = 2

= 2 cos^{2} (x) + 5 cos (x) + 2

= 2 cos^{2} (x) + 5 cos (x) + 2

2 + 2 cos^{2} (x) + 5 cos (x) = 0

Solve by substitution

2 + 2 cos^{2} (x) + 5 cos (x) = 0

Let:

cos (x) = u

2 + 2 u^{2} + 5 u = 0

2 + 2 u^{2} + 5 u = 0 : u = - \frac{1}{2}, u = - 2

2 + 2 u^{2} + 5 u = 0

Write in the standard form

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

2 u^{2} + 5 u + 2 = 0

Solve with the quadratic formula

2 u^{2} + 5 u + 2 = 0

Quadratic Equation Formula:

For

a = 2, b = 5, c = 2

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

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

5^{2} - 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

Subtract the numbers:

25 - 16 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 5 \pm 3}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 5 + 3}{2 \cdot 2}, u_{2} = \frac{- 5 - 3}{2 \cdot 2}

u = \frac{- 5 + 3}{2 \cdot 2} : - \frac{1}{2}

\frac{- 5 + 3}{2 \cdot 2}

Add/Subtract the numbers:

- 5 + 3 = - 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{- 5 - 3}{2 \cdot 2} : - 2

\frac{- 5 - 3}{2 \cdot 2}

Subtract the numbers:

- 5 - 3 = - 8

= \frac{- 8}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 8}{4}

Apply the fraction rule:

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

= - \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

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) = - 2 :

No Solution

cos (x) = - 2

- 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

Popular Examples

2cos(θ)=1

2 cos (θ) = 1

sin(x+pi/6)-sin(x-pi/6)= 1/2

sin (x + \frac{π}{6}) - sin (x - \frac{π}{6}) = \frac{1}{2}

tan(θ)= 5/12

tan (θ) = \frac{5}{12}

cos(3x)=(sqrt(3))/2

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

sin(x)=cos(x)-1

sin (x) = cos (x) - 1

Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)+5cos(x)+3=0 ?
The general solution for cos(2x)+5cos(x)+3=0 is x=(2pi)/3+2pin,x=(4pi)/3+2pin