What is the general solution for cos(x)(cos(x)+5)=2sin^2(x) ?

The general solution for cos(x)(cos(x)+5)=2sin^2(x) is x=1.23095…+2pin,x=2pi-1.23095…+2pin

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cos(x)(cos(x)+5)=2sin^2(x)

Solution

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

Solution

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Degrees

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 sin^{2} (x)

from both sides

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

Rewrite using trig identities

(5 + cos (x)) cos (x) - 2 sin^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

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

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

Simplify

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

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

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

Expand

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

cos (x) (5 + cos (x))

Apply the distributive law:

a (b + c) = ab + a c

a = cos (x), b = 5, c = cos (x)

= cos (x) \cdot 5 + cos (x) cos (x)

= 5 cos (x) + cos (x) cos (x)

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

cos (x) cos (x)

Apply exponent rule:

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

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 1, c = cos^{2} (x)

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

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 + 2 cos^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

cos^{2} (x) + 2 cos^{2} (x) = 3 cos^{2} (x)

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

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

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

- 2 + 3 cos^{2} (x) + 5 cos (x) = 0

Solve by substitution

- 2 + 3 cos^{2} (x) + 5 cos (x) = 0

Let:

cos (x) = u

- 2 + 3 u^{2} + 5 u = 0

- 2 + 3 u^{2} + 5 u = 0 : u = \frac{1}{3}, u = - 2

- 2 + 3 u^{2} + 5 u = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 3, b = 5, c = - 2

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

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

5^{2} - 4 \cdot 3 (- 2) = 7

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 5 + 7}{2 \cdot 3}, u_{2} = \frac{- 5 - 7}{2 \cdot 3}

u = \frac{- 5 + 7}{2 \cdot 3} : \frac{1}{3}

\frac{- 5 + 7}{2 \cdot 3}

Add/Subtract the numbers:

- 5 + 7 = 2

= \frac{2}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

u = \frac{- 5 - 7}{2 \cdot 3} : - 2

\frac{- 5 - 7}{2 \cdot 3}

Subtract the numbers:

- 5 - 7 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

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

cos (x) = \frac{1}{3}

Apply trig inverse properties

cos (x) = \frac{1}{3}

General solutions for

cos (x) = \frac{1}{3}

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

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = - 2 :

No Solution

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

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cot^{2} (x) = \frac{tan ( x )}{2}

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)(cos(x)+5)=2sin^2(x) ?
The general solution for cos(x)(cos(x)+5)=2sin^2(x) is x=1.23095…+2pin,x=2pi-1.23095…+2pin