What is the general solution for 2cos(x)+2sin(x)=sqrt(6) ?

The general solution for 2cos(x)+2sin(x)=sqrt(6) is x=0.26179…+2pin,x=1.30899…+2pin

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2cos(x)+2sin(x)=sqrt(6)

Solution

2 cos (x) + 2 sin (x) = 6

Solution

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

Degrees

x = 1 5^{\circ} + 36 0^{\circ} n, x = 7 5^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (x) + 2 sin (x) = 6

Subtract

2 sin (x)

from both sides

2 cos (x) = 6 - 2 sin (x)

Square both sides

(2 cos (x))^{2} = (6 - 2 sin (x))^{2}

Subtract

(6 - 2 sin (x))^{2}

from both sides

4 cos^{2} (x) - 6 + 46 sin (x) - 4 sin^{2} (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

- 6 + 4 (1 - sin^{2} (x)) - 4 sin^{2} (x) + 4 sin (x) 6 : 46 sin (x) - 8 sin^{2} (x) - 2

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

= - 6 + 4 (1 - sin^{2} (x)) - 4 sin^{2} (x) + 46 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)

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

Simplify

- 6 + 4 - 4 sin^{2} (x) - 4 sin^{2} (x) + 4 sin (x) 6 : 46 sin (x) - 8 sin^{2} (x) - 2

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

Add similar elements:

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

= - 6 + 4 - 8 sin^{2} (x) + 46 sin (x)

Add/Subtract the numbers:

- 6 + 4 = - 2

= 46 sin (x) - 8 sin^{2} (x) - 2

= 46 sin (x) - 8 sin^{2} (x) - 2

= 46 sin (x) - 8 sin^{2} (x) - 2

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Write in the standard form

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

- 8 u^{2} + 46 u - 2 = 0

Solve with the quadratic formula

- 8 u^{2} + 46 u - 2 = 0

Quadratic Equation Formula:

For

a = - 8, b = 46, c = - 2

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

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

(46)^{2} - 4 (- 8) (- 2) = 42

(46)^{2} - 4 (- 8) (- 2)

Apply rule

- (- a) = a

= (46)^{2} - 4 \cdot 8 \cdot 2

(46)^{2} = 4^{2} \cdot 6

(46)^{2}

Apply exponent rule:

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

= 4^{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

= 4^{2} \cdot 6

4 \cdot 8 \cdot 2 = 64

4 \cdot 8 \cdot 2

Multiply the numbers:

4 \cdot 8 \cdot 2 = 64

= 64

= 4^{2} \cdot 6 - 64

4^{2} \cdot 6 = 96

4^{2} \cdot 6

4^{2} = 16

= 16 \cdot 6

Multiply the numbers:

16 \cdot 6 = 96

= 96

= 96 - 64

Subtract the numbers:

96 - 64 = 32

= 32

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 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

= 2^{5}

= 2^{5}

Apply exponent rule:

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

= 2^{4} \cdot 2

Apply radical rule:

= 2 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

Refine

= 42

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

Separate the solutions

u_{1} = \frac{- 4 6 + 4 2}{2 ( - 8 )}, u_{2} = \frac{- 4 6 - 4 2}{2 ( - 8 )}

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

\frac{- 4 6 + 4 2}{2 ( - 8 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 6 + 4 2}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 4 6 + 4 2}{- 16}

Apply the fraction rule:

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

- 46 + 42 = - (46 - 42)

= \frac{4 6 - 4 2}{16}

Factor out common term

4

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

Cancel the common factor:

4

= \frac{6 - 2}{4}

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

\frac{- 4 6 - 4 2}{2 ( - 8 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 6 - 4 2}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 4 6 - 4 2}{- 16}

Apply the fraction rule:

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

- 46 - 42 = - (46 + 42)

= \frac{4 6 + 4 2}{16}

Factor out common term

4

= \frac{4 ( 6 + 2 )}{16}

Cancel the common factor:

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) + 2 sin (x) = 6

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

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) + 2 sin (x) = 6

plug in

x = arcsin (\frac{6 - 2}{4}) + 2 π 1

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

Refine

2.44948 \dots = 2.44948 \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) + 2 sin (x) = 6

plug in

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

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

Refine

- 1.41421 \dots = 2.44948 \dots

\Rightarrow False

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) + 2 sin (x) = 6

plug in

x = arcsin (\frac{6 + 2}{4}) + 2 π 1

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

Refine

2.44948 \dots = 2.44948 \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) + 2 sin (x) = 6

plug in

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

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

Refine

1.41421 \dots = 2.44948 \dots

\Rightarrow False

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

Show solutions in decimal form

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

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x)+2sin(x)=sqrt(6) ?
The general solution for 2cos(x)+2sin(x)=sqrt(6) is x=0.26179…+2pin,x=1.30899…+2pin