What is the general solution for 6cos(x)+6sin(x)=3sqrt(6) ?

The general solution for 6cos(x)+6sin(x)=3sqrt(6) is x=0.26179…+2pin,x=1.30899…+2pin

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Popular >

6cos(x)+6sin(x)=3sqrt(6)

Solution

6 cos (x) + 6 sin (x) = 36

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

6 cos (x) + 6 sin (x) = 36

Subtract

6 sin (x)

from both sides

6 cos (x) = 36 - 6 sin (x)

Square both sides

(6 cos (x))^{2} = (36 - 6 sin (x))^{2}

Subtract

(36 - 6 sin (x))^{2}

from both sides

36 cos^{2} (x) - 54 + 366 sin (x) - 36 sin^{2} (x) = 0

Rewrite using trig identities

- 54 + 36 cos^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6

Use the Pythagorean identity:

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

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

= - 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 36 sin (x) 6

Simplify

- 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 36 sin (x) 6 : 366 sin (x) - 72 sin^{2} (x) - 18

- 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 36 sin (x) 6

= - 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 366 sin (x)

Expand

36 (1 - sin^{2} (x)) : 36 - 36 sin^{2} (x)

36 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 36, b = 1, c = sin^{2} (x)

= 36 \cdot 1 - 36 sin^{2} (x)

Multiply the numbers:

36 \cdot 1 = 36

= 36 - 36 sin^{2} (x)

= - 54 + 36 - 36 sin^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6

Simplify

- 54 + 36 - 36 sin^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6 : 366 sin (x) - 72 sin^{2} (x) - 18

- 54 + 36 - 36 sin^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6

Add similar elements:

- 36 sin^{2} (x) - 36 sin^{2} (x) = - 72 sin^{2} (x)

= - 54 + 36 - 72 sin^{2} (x) + 366 sin (x)

Add/Subtract the numbers:

- 54 + 36 = - 18

= 366 sin (x) - 72 sin^{2} (x) - 18

= 366 sin (x) - 72 sin^{2} (x) - 18

= 366 sin (x) - 72 sin^{2} (x) - 18

- 18 - 72 sin^{2} (x) + 36 sin (x) 6 = 0

Solve by substitution

- 18 - 72 sin^{2} (x) + 36 sin (x) 6 = 0

Let:

sin (x) = u

- 18 - 72 u^{2} + 36 u 6 = 0

- 18 - 72 u^{2} + 36 u 6 = 0 : u = \frac{6 - 2}{4}, u = \frac{6 + 2}{4}

- 18 - 72 u^{2} + 36 u 6 = 0

Write in the standard form

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

- 72 u^{2} + 366 u - 18 = 0

Solve with the quadratic formula

- 72 u^{2} + 366 u - 18 = 0

Quadratic Equation Formula:

For

a = - 72, b = 366, c = - 18

u_{1, 2} = \frac{- 36 6 \pm ( 36 6 ) ^{2} - 4 ( - 72 ) ( - 18 )}{2 ( - 72 )}

u_{1, 2} = \frac{- 36 6 \pm ( 36 6 ) ^{2} - 4 ( - 72 ) ( - 18 )}{2 ( - 72 )}

(366)^{2} - 4 (- 72) (- 18) = 362

(366)^{2} - 4 (- 72) (- 18)

Apply rule

- (- a) = a

= (366)^{2} - 4 \cdot 72 \cdot 18

(366)^{2} = 3 6^{2} \cdot 6

(366)^{2}

Apply exponent rule:

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

= 3 6^{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

= 3 6^{2} \cdot 6

4 \cdot 72 \cdot 18 = 5184

4 \cdot 72 \cdot 18

Multiply the numbers:

4 \cdot 72 \cdot 18 = 5184

= 5184

= 3 6^{2} \cdot 6 - 5184

3 6^{2} \cdot 6 = 7776

3 6^{2} \cdot 6

3 6^{2} = 1296

= 1296 \cdot 6

Multiply the numbers:

1296 \cdot 6 = 7776

= 7776

= 7776 - 5184

Subtract the numbers:

7776 - 5184 = 2592

= 2592

Prime factorization of

2592 : 2^{5} \cdot 3^{4}

2592

2592

divides by

22592 = 1296 \cdot 2

= 2 \cdot 1296

1296

divides by

21296 = 648 \cdot 2

= 2 \cdot 2 \cdot 648

648

divides by

2648 = 324 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 324

324

divides by

2324 = 162 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 162

162

divides by

2162 = 81 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 81

81

divides by

381 = 27 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 27

27

divides by

327 = 9 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

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

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

Apply exponent rule:

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

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

Apply radical rule:

= 2 2^{4} 3^{4}

Apply radical rule:

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

= 2^{2} 2 3^{4}

Apply radical rule:

3^{4} = 3^{\frac{4}{2}} = 3^{2}

= 2^{2} \cdot 3^{2} 2

Refine

= 362

u_{1, 2} = \frac{- 36 6 \pm 36 2}{2 ( - 72 )}

Separate the solutions

u_{1} = \frac{- 36 6 + 36 2}{2 ( - 72 )}, u_{2} = \frac{- 36 6 - 36 2}{2 ( - 72 )}

u = \frac{- 36 6 + 36 2}{2 ( - 72 )} : \frac{6 - 2}{4}

\frac{- 36 6 + 36 2}{2 ( - 72 )}

Remove parentheses:

(- a) = - a

= \frac{- 36 6 + 36 2}{- 2 \cdot 72}

Multiply the numbers:

2 \cdot 72 = 144

= \frac{- 36 6 + 36 2}{- 144}

Apply the fraction rule:

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

- 366 + 362 = - (366 - 362)

= \frac{36 6 - 36 2}{144}

Factor out common term

36

= \frac{36 ( 6 - 2 )}{144}

Cancel the common factor:

36

= \frac{6 - 2}{4}

u = \frac{- 36 6 - 36 2}{2 ( - 72 )} : \frac{6 + 2}{4}

\frac{- 36 6 - 36 2}{2 ( - 72 )}

Remove parentheses:

(- a) = - a

= \frac{- 36 6 - 36 2}{- 2 \cdot 72}

Multiply the numbers:

2 \cdot 72 = 144

= \frac{- 36 6 - 36 2}{- 144}

Apply the fraction rule:

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

- 366 - 362 = - (366 + 362)

= \frac{36 6 + 36 2}{144}

Factor out common term

36

= \frac{36 ( 6 + 2 )}{144}

Cancel the common factor:

36

= \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

6 cos (x) + 6 sin (x) = 36

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

6 cos (x) + 6 sin (x) = 36

plug in

x = arcsin (\frac{6 - 2}{4}) + 2 π 1

6 cos (arcsin (\frac{6 - 2}{4}) + 2 π 1) + 6 sin (arcsin (\frac{6 - 2}{4}) + 2 π 1) = 36

Refine

7.34846 \dots = 7.34846 \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

6 cos (x) + 6 sin (x) = 36

plug in

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

6 cos (π - arcsin (\frac{6 - 2}{4}) + 2 π 1) + 6 sin (π - arcsin (\frac{6 - 2}{4}) + 2 π 1) = 36

Refine

- 4.24264 \dots = 7.34846 \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

6 cos (x) + 6 sin (x) = 36

plug in

x = arcsin (\frac{6 + 2}{4}) + 2 π 1

6 cos (arcsin (\frac{6 + 2}{4}) + 2 π 1) + 6 sin (arcsin (\frac{6 + 2}{4}) + 2 π 1) = 36

Refine

7.34846 \dots = 7.34846 \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

6 cos (x) + 6 sin (x) = 36

plug in

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

6 cos (π - arcsin (\frac{6 + 2}{4}) + 2 π 1) + 6 sin (π - arcsin (\frac{6 + 2}{4}) + 2 π 1) = 36

Refine

4.24264 \dots = 7.34846 \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 6cos(x)+6sin(x)=3sqrt(6) ?
The general solution for 6cos(x)+6sin(x)=3sqrt(6) is x=0.26179…+2pin,x=1.30899…+2pin