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

The general solution for cos^2(x)-3cos(x)-2=0 is x=2.16705…+2pin,x=-2.16705…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(x)-3cos(x)-2=0

Solution

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

Solution

x = 2.16705 \dots + 2 πn, x = - 2.16705 \dots + 2 πn

Degrees

x = 124.16325 \dots^{\circ} + 36 0^{\circ} n, x = - 124.16325 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

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

u^{2} - 3 u - 2 = 0 : u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = - 3, c = - 2

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

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

(- 3)^{2} - 4 \cdot 1 \cdot (- 2) = 17

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

Apply rule

- (- a) = a

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 3)^{2} = 3^{2}

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

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

Add the numbers:

9 + 8 = 17

= 17

u_{1, 2} = \frac{- ( - 3 ) \pm 17}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 17}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 17}{2 \cdot 1}

u = \frac{- ( - 3 ) + 17}{2 \cdot 1} : \frac{3 + 17}{2}

\frac{- ( - 3 ) + 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 + 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 + 17}{2}

u = \frac{- ( - 3 ) - 17}{2 \cdot 1} : \frac{3 - 17}{2}

\frac{- ( - 3 ) - 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 - 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 - 17}{2}

The solutions to the quadratic equation are:

u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

Substitute back

u = cos (x)

cos (x) = \frac{3 + 17}{2}, cos (x) = \frac{3 - 17}{2}

cos (x) = \frac{3 + 17}{2}, cos (x) = \frac{3 - 17}{2}

cos (x) = \frac{3 + 17}{2} :

No Solution

cos (x) = \frac{3 + 17}{2}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{3 - 17}{2} : x = arccos (\frac{3 - 17}{2}) + 2 πn, x = - arccos (\frac{3 - 17}{2}) + 2 πn

cos (x) = \frac{3 - 17}{2}

Apply trig inverse properties

cos (x) = \frac{3 - 17}{2}

General solutions for

cos (x) = \frac{3 - 17}{2}

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

x = arccos (\frac{3 - 17}{2}) + 2 πn, x = - arccos (\frac{3 - 17}{2}) + 2 πn

x = arccos (\frac{3 - 17}{2}) + 2 πn, x = - arccos (\frac{3 - 17}{2}) + 2 πn

Combine all the solutions

x = arccos (\frac{3 - 17}{2}) + 2 πn, x = - arccos (\frac{3 - 17}{2}) + 2 πn

Show solutions in decimal form

x = 2.16705 \dots + 2 πn, x = - 2.16705 \dots + 2 πn

Popular Examples

tan(4x)=tan(2x)

tan (4 x) = tan (2 x)

solvefor x,cos(2x+5x)=cos(Hx)solve for

x, cos (2 x + 5 x) = cos (H x)

solvefor x,-4cos(x)-4cos(2x)=0 solve for

x, - 4 cos (x) - 4 cos (2 x) = 0

3-2cos(x)=3-2sin(x)

3 - 2 cos (x) = 3 - 2 sin (x)

sqrt(3)sin(x)=sin(x)tan(x)

3 sin (x) = sin (x) tan (x)

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)-3cos(x)-2=0 ?
The general solution for cos^2(x)-3cos(x)-2=0 is x=2.16705…+2pin,x=-2.16705…+2pin