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

The general solution for 6sin(x)=cos(x)-3 is x=-2.46068…+2pin,x=2pi-0.35061…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

6sin(x)=cos(x)-3

Solution

6 sin (x) = cos (x) - 3

Solution

x = - 2.46068 \dots + 2 πn, x = 2 π - 0.35061 \dots + 2 πn

Degrees

x = - 140.98675 \dots^{\circ} + 36 0^{\circ} n, x = 339.91139 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 sin (x) = cos (x) - 3

Square both sides

(6 sin (x))^{2} = (cos (x) - 3)^{2}

Subtract

(cos (x) - 3)^{2}

from both sides

36 sin^{2} (x) - cos^{2} (x) + 6 cos (x) - 9 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= - 9 - cos^{2} (x) + 36 (1 - cos^{2} (x)) + 6 cos (x)

Simplify

- 9 - cos^{2} (x) + 36 (1 - cos^{2} (x)) + 6 cos (x) : 6 cos (x) - 37 cos^{2} (x) + 27

- 9 - cos^{2} (x) + 36 (1 - cos^{2} (x)) + 6 cos (x)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

36 \cdot 1 = 36

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

= - 9 - cos^{2} (x) + 36 - 36 cos^{2} (x) + 6 cos (x)

Simplify

- 9 - cos^{2} (x) + 36 - 36 cos^{2} (x) + 6 cos (x) : 6 cos (x) - 37 cos^{2} (x) + 27

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

Group like terms

= - cos^{2} (x) - 36 cos^{2} (x) + 6 cos (x) - 9 + 36

Add similar elements:

- cos^{2} (x) - 36 cos^{2} (x) = - 37 cos^{2} (x)

= - 37 cos^{2} (x) + 6 cos (x) - 9 + 36

Add/Subtract the numbers:

- 9 + 36 = 27

= 6 cos (x) - 37 cos^{2} (x) + 27

= 6 cos (x) - 37 cos^{2} (x) + 27

= 6 cos (x) - 37 cos^{2} (x) + 27

27 - 37 cos^{2} (x) + 6 cos (x) = 0

Solve by substitution

27 - 37 cos^{2} (x) + 6 cos (x) = 0

Let:

cos (x) = u

27 - 37 u^{2} + 6 u = 0

27 - 37 u^{2} + 6 u = 0 : u = - \frac{3 ( 4 7 - 1 )}{37}, u = \frac{3 ( 1 + 4 7 )}{37}

27 - 37 u^{2} + 6 u = 0

Write in the standard form

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

- 37 u^{2} + 6 u + 27 = 0

Solve with the quadratic formula

- 37 u^{2} + 6 u + 27 = 0

Quadratic Equation Formula:

For

a = - 37, b = 6, c = 27

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 37 ) \cdot 27}{2 ( - 37 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 37 ) \cdot 27}{2 ( - 37 )}

6^{2} - 4 (- 37) \cdot 27 = 247

6^{2} - 4 (- 37) \cdot 27

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 37 \cdot 27

Multiply the numbers:

4 \cdot 37 \cdot 27 = 3996

= 6^{2} + 3996

6^{2} = 36

= 36 + 3996

Add the numbers:

36 + 3996 = 4032

= 4032

Prime factorization of

4032 : 2^{6} \cdot 3^{2} \cdot 7

4032

4032

divides by

24032 = 2016 \cdot 2

= 2 \cdot 2016

2016

divides by

22016 = 1008 \cdot 2

= 2 \cdot 2 \cdot 1008

1008

divides by

21008 = 504 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 504

504

divides by

2504 = 252 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 252

252

divides by

2252 = 126 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 126

126

divides by

2126 = 63 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 63

63

divides by

363 = 21 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 21

21

divides by

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7

2, 3, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7

= 2^{6} \cdot 3^{2} \cdot 7

= 2^{6} \cdot 3^{2} \cdot 7

Apply radical rule:

= 7 2^{6} 3^{2}

Apply radical rule:

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

= 2^{3} 7 3^{2}

Apply radical rule:

3^{2} = 3

= 2^{3} \cdot 37

Refine

= 247

u_{1, 2} = \frac{- 6 \pm 24 7}{2 ( - 37 )}

Separate the solutions

u_{1} = \frac{- 6 + 24 7}{2 ( - 37 )}, u_{2} = \frac{- 6 - 24 7}{2 ( - 37 )}

u = \frac{- 6 + 24 7}{2 ( - 37 )} : - \frac{3 ( 4 7 - 1 )}{37}

\frac{- 6 + 24 7}{2 ( - 37 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 24 7}{- 2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 6 + 24 7}{- 74}

Apply the fraction rule:

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

= - \frac{- 6 + 24 7}{74}

Cancel

\frac{- 6 + 24 7}{74} : \frac{3 ( 4 7 - 1 )}{37}

\frac{- 6 + 24 7}{74}

Factor

- 6 + 247 : 6 (- 1 + 47)

- 6 + 247

Rewrite as

= - 6 \cdot 1 + 6 \cdot 47

Factor out common term

6

= 6 (- 1 + 47)

= \frac{6 ( - 1 + 4 7 )}{74}

Cancel the common factor:

2

= \frac{3 ( 4 7 - 1 )}{37}

= - \frac{3 ( 4 7 - 1 )}{37}

u = \frac{- 6 - 24 7}{2 ( - 37 )} : \frac{3 ( 1 + 4 7 )}{37}

\frac{- 6 - 24 7}{2 ( - 37 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 24 7}{- 2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 6 - 24 7}{- 74}

Apply the fraction rule:

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

- 6 - 247 = - (6 + 247)

= \frac{6 + 24 7}{74}

Factor

6 + 247 : 6 (1 + 47)

6 + 247

Rewrite as

= 6 \cdot 1 + 6 \cdot 47

Factor out common term

6

= 6 (1 + 47)

= \frac{6 ( 1 + 4 7 )}{74}

Cancel the common factor:

2

= \frac{3 ( 1 + 4 7 )}{37}

The solutions to the quadratic equation are:

u = - \frac{3 ( 4 7 - 1 )}{37}, u = \frac{3 ( 1 + 4 7 )}{37}

Substitute back

u = cos (x)

cos (x) = - \frac{3 ( 4 7 - 1 )}{37}, cos (x) = \frac{3 ( 1 + 4 7 )}{37}

cos (x) = - \frac{3 ( 4 7 - 1 )}{37}, cos (x) = \frac{3 ( 1 + 4 7 )}{37}

cos (x) = - \frac{3 ( 4 7 - 1 )}{37} : x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn

cos (x) = - \frac{3 ( 4 7 - 1 )}{37}

Apply trig inverse properties

cos (x) = - \frac{3 ( 4 7 - 1 )}{37}

General solutions for

cos (x) = - \frac{3 ( 4 7 - 1 )}{37}

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

x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn

x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn

cos (x) = \frac{3 ( 1 + 4 7 )}{37} : x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

cos (x) = \frac{3 ( 1 + 4 7 )}{37}

Apply trig inverse properties

cos (x) = \frac{3 ( 1 + 4 7 )}{37}

General solutions for

cos (x) = \frac{3 ( 1 + 4 7 )}{37}

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

x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Combine all the solutions

x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

6 sin (x) = cos (x) - 3

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

Check the solution

arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn :

False

arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn

Plug in

n = 1

arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1

For

6 sin (x) = cos (x) - 3

plug in

x = arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1

6 sin (arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1) = cos (arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1) - 3

Refine

3.77700 \dots = - 3.77700 \dots

\Rightarrow False

Check the solution

- arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn :

True

- arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn

Plug in

n = 1

- arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1

For

6 sin (x) = cos (x) - 3

plug in

x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1

6 sin (- arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1) = cos (- arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 π 1) - 3

Refine

- 3.77700 \dots = - 3.77700 \dots

\Rightarrow True

Check the solution

arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn :

False

arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Plug in

n = 1

arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1

For

6 sin (x) = cos (x) - 3

plug in

x = arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1

6 sin (arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1) = cos (arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1) - 3

Refine

2.06083 \dots = - 2.06083 \dots

\Rightarrow False

Check the solution

2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn :

True

2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1

For

6 sin (x) = cos (x) - 3

plug in

x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1

6 sin (2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1) = cos (2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 π 1) - 3

Refine

- 2.06083 \dots = - 2.06083 \dots

\Rightarrow True

x = - arccos (- \frac{3 ( 4 7 - 1 )}{37}) + 2 πn, x = 2 π - arccos (\frac{3 ( 1 + 4 7 )}{37}) + 2 πn

Show solutions in decimal form

x = - 2.46068 \dots + 2 πn, x = 2 π - 0.35061 \dots + 2 πn

Popular Examples

-2sin(4x)=0

- 2 sin (4 x) = 0

2sin(x)+sqrt(3)=0,0<= x<= 2pi

2 sin (x) + 3 = 0, 0 \leq x \leq 2 π

cos(2x)=1-sin(2x),0<= x<= 2pi

cos (2 x) = 1 - sin (2 x), 0 \leq x \leq 2 π

0=1+2sin(θ)

0 = 1 + 2 sin (θ)

csc(x-pi/3)=2

csc (x - \frac{π}{3}) = 2

Frequently Asked Questions (FAQ)

What is the general solution for 6sin(x)=cos(x)-3 ?
The general solution for 6sin(x)=cos(x)-3 is x=-2.46068…+2pin,x=2pi-0.35061…+2pin