What is the general solution for 2cos(x)-2sqrt(3)*sin(x)=sqrt(8) ?

The general solution for 2cos(x)-2sqrt(3)*sin(x)=sqrt(8) is x=pi+1.30899…+2pin,x=-0.26179…+2pin

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2cos(x)-2sqrt(3)*sin(x)=sqrt(8)

Solution

2 cos (x) - 23 \cdot sin (x) = 8

Solution

x = π + 1.30899 \dots + 2 πn, x = - 0.26179 \dots + 2 πn

Degrees

x = 25 5^{\circ} + 36 0^{\circ} n, x = - 1 5^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (x) - 23 sin (x) = 8

Add

23 sin (x)

to both sides

2 cos (x) = 22 + 23 sin (x)

Square both sides

(2 cos (x))^{2} = (22 + 23 sin (x))^{2}

Subtract

(22 + 23 sin (x))^{2}

from both sides

4 cos^{2} (x) - 8 - 86 sin (x) - 12 sin^{2} (x) = 0

Rewrite using trig identities

- 8 - 12 sin^{2} (x) + 4 cos^{2} (x) - 8 sin (x) 6

Use the Pythagorean identity:

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

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

= - 8 - 12 sin^{2} (x) + 4 (1 - sin^{2} (x)) - 8 sin (x) 6

Simplify

- 8 - 12 sin^{2} (x) + 4 (1 - sin^{2} (x)) - 8 sin (x) 6 : - 16 sin^{2} (x) - 86 sin (x) - 4

- 8 - 12 sin^{2} (x) + 4 (1 - sin^{2} (x)) - 8 sin (x) 6

= - 8 - 12 sin^{2} (x) + 4 (1 - sin^{2} (x)) - 86 sin (x)

Expand

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

= 4 \cdot 1 - 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (x)

= - 8 - 12 sin^{2} (x) + 4 - 4 sin^{2} (x) - 8 sin (x) 6

Simplify

- 8 - 12 sin^{2} (x) + 4 - 4 sin^{2} (x) - 8 sin (x) 6 : - 16 sin^{2} (x) - 86 sin (x) - 4

- 8 - 12 sin^{2} (x) + 4 - 4 sin^{2} (x) - 8 sin (x) 6

Group like terms

= - 12 sin^{2} (x) - 4 sin^{2} (x) - 86 sin (x) - 8 + 4

Add similar elements:

- 12 sin^{2} (x) - 4 sin^{2} (x) = - 16 sin^{2} (x)

= - 16 sin^{2} (x) - 86 sin (x) - 8 + 4

Add/Subtract the numbers:

- 8 + 4 = - 4

= - 16 sin^{2} (x) - 86 sin (x) - 4

= - 16 sin^{2} (x) - 86 sin (x) - 4

= - 16 sin^{2} (x) - 86 sin (x) - 4

- 4 - 16 sin^{2} (x) - 8 sin (x) 6 = 0

Solve by substitution

- 4 - 16 sin^{2} (x) - 8 sin (x) 6 = 0

Let:

sin (x) = u

- 4 - 16 u^{2} - 8 u 6 = 0

- 4 - 16 u^{2} - 8 u 6 = 0 : u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

- 4 - 16 u^{2} - 8 u 6 = 0

Write in the standard form

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

- 16 u^{2} - 86 u - 4 = 0

Solve with the quadratic formula

- 16 u^{2} - 86 u - 4 = 0

Quadratic Equation Formula:

For

a = - 16, b = - 86, c = - 4

u_{1, 2} = \frac{- ( - 8 6 ) \pm ( - 8 6 ) ^{2} - 4 ( - 16 ) ( - 4 )}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 8 6 ) \pm ( - 8 6 ) ^{2} - 4 ( - 16 ) ( - 4 )}{2 ( - 16 )}

(- 86)^{2} - 4 (- 16) (- 4) = 82

(- 86)^{2} - 4 (- 16) (- 4)

Apply rule

- (- a) = a

= (- 86)^{2} - 4 \cdot 16 \cdot 4

(- 86)^{2} = 8^{2} \cdot 6

(- 86)^{2}

Apply exponent rule:

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

n

is even

(- 86)^{2} = (86)^{2}

= (86)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 8^{2} (6)^{2}

(6)^{2} : 6

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 6

= 8^{2} \cdot 6

4 \cdot 16 \cdot 4 = 256

4 \cdot 16 \cdot 4

Multiply the numbers:

4 \cdot 16 \cdot 4 = 256

= 256

= 8^{2} \cdot 6 - 256

8^{2} \cdot 6 = 384

8^{2} \cdot 6

8^{2} = 64

= 64 \cdot 6

Multiply the numbers:

64 \cdot 6 = 384

= 384

= 384 - 256

Subtract the numbers:

384 - 256 = 128

= 128

Prime factorization of

128 : 2^{7}

128

128

divides by

2128 = 64 \cdot 2

= 2 \cdot 64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{7}

= 2^{7}

Apply exponent rule:

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

= 2^{6} \cdot 2

Apply radical rule:

= 2 2^{6}

Apply radical rule:

2^{6} = 2^{\frac{6}{2}} = 2^{3}

= 2^{3} 2

Refine

= 82

u_{1, 2} = \frac{- ( - 8 6 ) \pm 8 2}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- ( - 8 6 ) + 8 2}{2 ( - 16 )}, u_{2} = \frac{- ( - 8 6 ) - 8 2}{2 ( - 16 )}

u = \frac{- ( - 8 6 ) + 8 2}{2 ( - 16 )} : - \frac{6 + 2}{4}

\frac{- ( - 8 6 ) + 8 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{8 6 + 8 2}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8 6 + 8 2}{- 32}

Apply the fraction rule:

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

= - \frac{8 6 + 8 2}{32}

Cancel

\frac{8 6 + 8 2}{32} : \frac{6 + 2}{4}

\frac{8 6 + 8 2}{32}

Factor out common term

8

= \frac{8 ( 6 + 2 )}{32}

Cancel the common factor:

8

= \frac{6 + 2}{4}

= - \frac{6 + 2}{4}

u = \frac{- ( - 8 6 ) - 8 2}{2 ( - 16 )} : - \frac{6 - 2}{4}

\frac{- ( - 8 6 ) - 8 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{8 6 - 8 2}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8 6 - 8 2}{- 32}

Apply the fraction rule:

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

= - \frac{8 6 - 8 2}{32}

Cancel

\frac{8 6 - 8 2}{32} : \frac{6 - 2}{4}

\frac{8 6 - 8 2}{32}

Factor out common term

8

= \frac{8 ( 6 - 2 )}{32}

Cancel the common factor:

8

= \frac{6 - 2}{4}

= - \frac{6 - 2}{4}

The solutions to the quadratic equation are:

u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

Substitute back

u = sin (x)

sin (x) = - \frac{6 + 2}{4}, sin (x) = - \frac{6 - 2}{4}

sin (x) = - \frac{6 + 2}{4}, sin (x) = - \frac{6 - 2}{4}

sin (x) = - \frac{6 + 2}{4} : x = arcsin (- \frac{6 + 2}{4}) + 2 πn, x = π + arcsin (\frac{6 + 2}{4}) + 2 πn

sin (x) = - \frac{6 + 2}{4}

Apply trig inverse properties

sin (x) = - \frac{6 + 2}{4}

General solutions for

sin (x) = - \frac{6 + 2}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{6 + 2}{4}) + 2 πn, x = π + arcsin (\frac{6 + 2}{4}) + 2 πn

x = arcsin (- \frac{6 + 2}{4}) + 2 πn, x = π + arcsin (\frac{6 + 2}{4}) + 2 πn

sin (x) = - \frac{6 - 2}{4} : x = arcsin (- \frac{6 - 2}{4}) + 2 πn, x = π + arcsin (\frac{6 - 2}{4}) + 2 πn

sin (x) = - \frac{6 - 2}{4}

Apply trig inverse properties

sin (x) = - \frac{6 - 2}{4}

General solutions for

sin (x) = - \frac{6 - 2}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{6 - 2}{4}) + 2 πn, x = π + arcsin (\frac{6 - 2}{4}) + 2 πn

x = arcsin (- \frac{6 - 2}{4}) + 2 πn, x = π + arcsin (\frac{6 - 2}{4}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{6 + 2}{4}) + 2 πn, x = π + arcsin (\frac{6 + 2}{4}) + 2 πn, x = arcsin (- \frac{6 - 2}{4}) + 2 πn, x = π + arcsin (\frac{6 - 2}{4}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

2 cos (x) - 23 sin (x) = 8

Remove the ones that don't agree with the equation.

Check the solution

arcsin (- \frac{6 + 2}{4}) + 2 πn :

False

arcsin (- \frac{6 + 2}{4}) + 2 πn

Plug in

n = 1

arcsin (- \frac{6 + 2}{4}) + 2 π 1

For

2 cos (x) - 23 sin (x) = 8

plug in

x = arcsin (- \frac{6 + 2}{4}) + 2 π 1

2 cos (arcsin (- \frac{6 + 2}{4}) + 2 π 1) - 23 sin (arcsin (- \frac{6 + 2}{4}) + 2 π 1) = 8

Refine

3.86370 \dots = 2.82842 \dots

\Rightarrow False

Check the solution

π + arcsin (\frac{6 + 2}{4}) + 2 πn :

True

π + arcsin (\frac{6 + 2}{4}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{6 + 2}{4}) + 2 π 1

For

2 cos (x) - 23 sin (x) = 8

plug in

x = π + arcsin (\frac{6 + 2}{4}) + 2 π 1

2 cos (π + arcsin (\frac{6 + 2}{4}) + 2 π 1) - 23 sin (π + arcsin (\frac{6 + 2}{4}) + 2 π 1) = 8

Refine

2.82842 \dots = 2.82842 \dots

\Rightarrow True

Check the solution

arcsin (- \frac{6 - 2}{4}) + 2 πn :

True

arcsin (- \frac{6 - 2}{4}) + 2 πn

Plug in

n = 1

arcsin (- \frac{6 - 2}{4}) + 2 π 1

For

2 cos (x) - 23 sin (x) = 8

plug in

x = arcsin (- \frac{6 - 2}{4}) + 2 π 1

2 cos (arcsin (- \frac{6 - 2}{4}) + 2 π 1) - 23 sin (arcsin (- \frac{6 - 2}{4}) + 2 π 1) = 8

Refine

2.82842 \dots = 2.82842 \dots

\Rightarrow True

Check the solution

π + arcsin (\frac{6 - 2}{4}) + 2 πn :

False

π + arcsin (\frac{6 - 2}{4}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{6 - 2}{4}) + 2 π 1

For

2 cos (x) - 23 sin (x) = 8

plug in

x = π + arcsin (\frac{6 - 2}{4}) + 2 π 1

2 cos (π + arcsin (\frac{6 - 2}{4}) + 2 π 1) - 23 sin (π + arcsin (\frac{6 - 2}{4}) + 2 π 1) = 8

Refine

- 1.03527 \dots = 2.82842 \dots

\Rightarrow False

x = π + arcsin (\frac{6 + 2}{4}) + 2 πn, x = arcsin (- \frac{6 - 2}{4}) + 2 πn

Show solutions in decimal form

x = π + 1.30899 \dots + 2 πn, x = - 0.26179 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x)-2sqrt(3)*sin(x)=sqrt(8) ?
The general solution for 2cos(x)-2sqrt(3)*sin(x)=sqrt(8) is x=pi+1.30899…+2pin,x=-0.26179…+2pin