What is the general solution for cos^2(x)-cos(x)=0.75 ?

The general solution for cos^2(x)-cos(x)=0.75 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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Popular Trigonometry >

cos^2(x)-cos(x)=0.75

Solution

cos^{2} (x) - cos (x) = 0.75

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) - cos (x) = 0.75

Solve by substitution

cos^{2} (x) - cos (x) = 0.75

Let:

cos (x) = u

u^{2} - u = 0.75

u^{2} - u = 0.75 : u = \frac{3}{2}, u = - \frac{1}{2}

u^{2} - u = 0.75

Multiply both sides by

100

u^{2} - u = 0.75

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

u^{2} \cdot 100 - u \cdot 100 = 0.75 \cdot 100

Refine

u^{2} \cdot 100 - u \cdot 100 = 75

u^{2} \cdot 100 - u \cdot 100 = 75

Move

75

to the left side

u^{2} \cdot 100 - u \cdot 100 = 75

Subtract

75

from both sides

u^{2} \cdot 100 - u \cdot 100 - 75 = 75 - 75

Simplify

u^{2} \cdot 100 - u \cdot 100 - 75 = 0

u^{2} \cdot 100 - u \cdot 100 - 75 = 0

Write in the standard form

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

100 u^{2} - 100 u - 75 = 0

Solve with the quadratic formula

100 u^{2} - 100 u - 75 = 0

Quadratic Equation Formula:

For

a = 100, b = - 100, c = - 75

u_{1, 2} = \frac{- ( - 100 ) \pm ( - 100 ) ^{2} - 4 \cdot 100 ( - 75 )}{2 \cdot 100}

u_{1, 2} = \frac{- ( - 100 ) \pm ( - 100 ) ^{2} - 4 \cdot 100 ( - 75 )}{2 \cdot 100}

(- 100)^{2} - 4 \cdot 100 (- 75) = 200

(- 100)^{2} - 4 \cdot 100 (- 75)

Apply rule

- (- a) = a

= (- 100)^{2} + 4 \cdot 100 \cdot 75

Apply exponent rule:

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

n

is even

(- 100)^{2} = 10 0^{2}

= 10 0^{2} + 4 \cdot 100 \cdot 75

Multiply the numbers:

4 \cdot 100 \cdot 75 = 30000

= 10 0^{2} + 30000

10 0^{2} = 10000

= 10000 + 30000

Add the numbers:

10000 + 30000 = 40000

= 40000

Factor the number:

40000 = 20 0^{2}

= 20 0^{2}

Apply radical rule:

20 0^{2} = 200

= 200

u_{1, 2} = \frac{- ( - 100 ) \pm 200}{2 \cdot 100}

Separate the solutions

u_{1} = \frac{- ( - 100 ) + 200}{2 \cdot 100}, u_{2} = \frac{- ( - 100 ) - 200}{2 \cdot 100}

u = \frac{- ( - 100 ) + 200}{2 \cdot 100} : \frac{3}{2}

\frac{- ( - 100 ) + 200}{2 \cdot 100}

Apply rule

- (- a) = a

= \frac{100 + 200}{2 \cdot 100}

Add the numbers:

100 + 200 = 300

= \frac{300}{2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{300}{200}

Cancel the common factor:

100

= \frac{3}{2}

u = \frac{- ( - 100 ) - 200}{2 \cdot 100} : - \frac{1}{2}

\frac{- ( - 100 ) - 200}{2 \cdot 100}

Apply rule

- (- a) = a

= \frac{100 - 200}{2 \cdot 100}

Subtract the numbers:

100 - 200 = - 100

= \frac{- 100}{2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{- 100}{200}

Apply the fraction rule:

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

= - \frac{100}{200}

Cancel the common factor:

100

= - \frac{1}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

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

No Solution

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

- 1 \leq cos (x) \leq 1

No Solution

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

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)-cos(x)=0.75 ?
The general solution for cos^2(x)-cos(x)=0.75 is x=(2pi)/3+2pin,x=(4pi)/3+2pin