What is the general solution for -4cos^2(x)+3cos(x)+1=0 ?

The general solution for -4cos^2(x)+3cos(x)+1=0 is x=1.82347…+2pin,x=-1.82347…+2pin,x=2pin

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Popular Trigonometry >

-4cos^2(x)+3cos(x)+1=0

Solution

- 4 cos^{2} (x) + 3 cos (x) + 1 = 0

Solution

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn, x = 2 πn

Degrees

x = 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = - 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n

Solution steps

- 4 cos^{2} (x) + 3 cos (x) + 1 = 0

Solve by substitution

- 4 cos^{2} (x) + 3 cos (x) + 1 = 0

Let:

cos (x) = u

- 4 u^{2} + 3 u + 1 = 0

- 4 u^{2} + 3 u + 1 = 0 : u = - \frac{1}{4}, u = 1

- 4 u^{2} + 3 u + 1 = 0

Solve with the quadratic formula

- 4 u^{2} + 3 u + 1 = 0

Quadratic Equation Formula:

For

a = - 4, b = 3, c = 1

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

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

3^{2} - 4 (- 4) \cdot 1 = 5

3^{2} - 4 (- 4) \cdot 1

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 4 \cdot 1

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Add the numbers:

9 + 16 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 ( - 4 )}, u_{2} = \frac{- 3 - 5}{2 ( - 4 )}

u = \frac{- 3 + 5}{2 ( - 4 )} : - \frac{1}{4}

\frac{- 3 + 5}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 + 5}{- 2 \cdot 4}

Add/Subtract the numbers:

- 3 + 5 = 2

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{2}{- 8}

Apply the fraction rule:

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

= - \frac{2}{8}

Cancel the common factor:

2

= - \frac{1}{4}

u = \frac{- 3 - 5}{2 ( - 4 )} : 1

\frac{- 3 - 5}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

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

Subtract the numbers:

- 3 - 5 = - 8

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 8}{- 8}

Apply the fraction rule:

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

= \frac{8}{8}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - \frac{1}{4}, u = 1

Substitute back

u = cos (x)

cos (x) = - \frac{1}{4}, cos (x) = 1

cos (x) = - \frac{1}{4}, cos (x) = 1

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

cos (x) = - \frac{1}{4}

Apply trig inverse properties

cos (x) = - \frac{1}{4}

General solutions for

cos (x) = - \frac{1}{4}

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

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn, x = 2 πn

Show solutions in decimal form

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn, x = 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -4cos^2(x)+3cos(x)+1=0 ?
The general solution for -4cos^2(x)+3cos(x)+1=0 is x=1.82347…+2pin,x=-1.82347…+2pin,x=2pin