What is the general solution for 4(cos(x)+1)cos(x)=3 ?

The general solution for 4(cos(x)+1)cos(x)=3 is x= pi/3+2pin,x=(5pi)/3+2pin

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4(cos(x)+1)cos(x)=3

Solution

4 (cos (x) + 1) cos (x) = 3

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

4 (cos (x) + 1) cos (x) = 3

Solve by substitution

4 (cos (x) + 1) cos (x) = 3

Let:

cos (x) = u

4 (u + 1) u = 3

4 (u + 1) u = 3 : u = \frac{1}{2}, u = - \frac{3}{2}

4 (u + 1) u = 3

Expand

4 (u + 1) u : 4 u^{2} + 4 u

4 (u + 1) u

= 4 u (u + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 4 u, b = u, c = 1

= 4 uu + 4 u \cdot 1

= 4 uu + 4 \cdot 1 \cdot u

Simplify

4 uu + 4 \cdot 1 \cdot u : 4 u^{2} + 4 u

4 uu + 4 \cdot 1 \cdot u

4 uu = 4 u^{2}

4 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 4 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 4 u^{2}

4 \cdot 1 \cdot u = 4 u

4 \cdot 1 \cdot u

Multiply the numbers:

4 \cdot 1 = 4

= 4 u

= 4 u^{2} + 4 u

= 4 u^{2} + 4 u

4 u^{2} + 4 u = 3

Move

3

to the left side

4 u^{2} + 4 u = 3

Subtract

3

from both sides

4 u^{2} + 4 u - 3 = 3 - 3

Simplify

4 u^{2} + 4 u - 3 = 0

4 u^{2} + 4 u - 3 = 0

Solve with the quadratic formula

4 u^{2} + 4 u - 3 = 0

Quadratic Equation Formula:

For

a = 4, b = 4, c = - 3

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

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

4^{2} - 4 \cdot 4 (- 3) = 8

4^{2} - 4 \cdot 4 (- 3)

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

Add the numbers:

16 + 48 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- 4 \pm 8}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 4 + 8}{2 \cdot 4}, u_{2} = \frac{- 4 - 8}{2 \cdot 4}

u = \frac{- 4 + 8}{2 \cdot 4} : \frac{1}{2}

\frac{- 4 + 8}{2 \cdot 4}

Add/Subtract the numbers:

- 4 + 8 = 4

= \frac{4}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{- 4 - 8}{2 \cdot 4} : - \frac{3}{2}

\frac{- 4 - 8}{2 \cdot 4}

Subtract the numbers:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 12}{8}

Apply the fraction rule:

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

= - \frac{12}{8}

Cancel the common factor:

4

= - \frac{3}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

No Solution

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

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4(cos(x)+1)cos(x)=3 ?
The general solution for 4(cos(x)+1)cos(x)=3 is x= pi/3+2pin,x=(5pi)/3+2pin