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

The general solution for 6sin(x)=cos(x)-2 is x=-2.64141…+2pin,x=2pi-0.16988…+2pin

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Popular Trigonometry >

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

Solution

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

Solution

x = - 2.64141 \dots + 2 πn, x = 2 π - 0.16988 \dots + 2 πn

Degrees

x = - 151.34184 \dots^{\circ} + 36 0^{\circ} n, x = 350.26648 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Square both sides

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

Subtract

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

from both sides

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

Rewrite using trig identities

- 4 - cos^{2} (x) + 36 sin^{2} (x) + 4 cos (x)

Use the Pythagorean identity:

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

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

= - 4 - cos^{2} (x) + 36 (1 - cos^{2} (x)) + 4 cos (x)

Simplify

- 4 - cos^{2} (x) + 36 (1 - cos^{2} (x)) + 4 cos (x) : 4 cos (x) - 37 cos^{2} (x) + 32

- 4 - cos^{2} (x) + 36 (1 - cos^{2} (x)) + 4 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)

= - 4 - cos^{2} (x) + 36 - 36 cos^{2} (x) + 4 cos (x)

Simplify

- 4 - cos^{2} (x) + 36 - 36 cos^{2} (x) + 4 cos (x) : 4 cos (x) - 37 cos^{2} (x) + 32

- 4 - cos^{2} (x) + 36 - 36 cos^{2} (x) + 4 cos (x)

Group like terms

= - cos^{2} (x) - 36 cos^{2} (x) + 4 cos (x) - 4 + 36

Add similar elements:

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

= - 37 cos^{2} (x) + 4 cos (x) - 4 + 36

Add/Subtract the numbers:

- 4 + 36 = 32

= 4 cos (x) - 37 cos^{2} (x) + 32

= 4 cos (x) - 37 cos^{2} (x) + 32

= 4 cos (x) - 37 cos^{2} (x) + 32

32 - 37 cos^{2} (x) + 4 cos (x) = 0

Solve by substitution

32 - 37 cos^{2} (x) + 4 cos (x) = 0

Let:

cos (x) = u

32 - 37 u^{2} + 4 u = 0

32 - 37 u^{2} + 4 u = 0 : u = - \frac{2 ( 3 33 - 1 )}{37}, u = \frac{2 ( 1 + 3 33 )}{37}

32 - 37 u^{2} + 4 u = 0

Write in the standard form

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

- 37 u^{2} + 4 u + 32 = 0

Solve with the quadratic formula

- 37 u^{2} + 4 u + 32 = 0

Quadratic Equation Formula:

For

a = - 37, b = 4, c = 32

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 37 ) \cdot 32}{2 ( - 37 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 37 ) \cdot 32}{2 ( - 37 )}

4^{2} - 4 (- 37) \cdot 32 = 1233

4^{2} - 4 (- 37) \cdot 32

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 37 \cdot 32

Multiply the numbers:

4 \cdot 37 \cdot 32 = 4736

= 4^{2} + 4736

4^{2} = 16

= 16 + 4736

Add the numbers:

16 + 4736 = 4752

= 4752

Prime factorization of

4752 : 2^{4} \cdot 3^{3} \cdot 11

4752

4752

divides by

24752 = 2376 \cdot 2

= 2 \cdot 2376

2376

divides by

22376 = 1188 \cdot 2

= 2 \cdot 2 \cdot 1188

1188

divides by

21188 = 594 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 594

594

divides by

2594 = 297 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 297

297

divides by

3297 = 99 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 99

99

divides by

399 = 33 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 33

33

divides by

333 = 11 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 11

2, 3, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 11

= 2^{4} \cdot 3^{3} \cdot 11

= 2^{4} \cdot 3^{3} \cdot 11

Apply exponent rule:

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

= 2^{4} \cdot 3^{2} \cdot 3 \cdot 11

Apply radical rule:

= 2^{4} 3^{2} 3 \cdot 11

Apply radical rule:

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

= 2^{2} 3^{2} 3 \cdot 11

Apply radical rule:

3^{2} = 3

= 2^{2} \cdot 3 3 \cdot 11

Refine

= 1233

u_{1, 2} = \frac{- 4 \pm 12 33}{2 ( - 37 )}

Separate the solutions

u_{1} = \frac{- 4 + 12 33}{2 ( - 37 )}, u_{2} = \frac{- 4 - 12 33}{2 ( - 37 )}

u = \frac{- 4 + 12 33}{2 ( - 37 )} : - \frac{2 ( 3 33 - 1 )}{37}

\frac{- 4 + 12 33}{2 ( - 37 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 + 12 33}{- 2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 4 + 12 33}{- 74}

Apply the fraction rule:

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

= - \frac{- 4 + 12 33}{74}

Cancel

\frac{- 4 + 12 33}{74} : \frac{2 ( 3 33 - 1 )}{37}

\frac{- 4 + 12 33}{74}

Factor

- 4 + 1233 : 4 (- 1 + 333)

- 4 + 1233

Rewrite as

= - 4 \cdot 1 + 4 \cdot 333

Factor out common term

4

= 4 (- 1 + 333)

= \frac{4 ( - 1 + 3 33 )}{74}

Cancel the common factor:

2

= \frac{2 ( 3 33 - 1 )}{37}

= - \frac{2 ( 3 33 - 1 )}{37}

u = \frac{- 4 - 12 33}{2 ( - 37 )} : \frac{2 ( 1 + 3 33 )}{37}

\frac{- 4 - 12 33}{2 ( - 37 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 - 12 33}{- 2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 4 - 12 33}{- 74}

Apply the fraction rule:

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

- 4 - 1233 = - (4 + 1233)

= \frac{4 + 12 33}{74}

Factor

4 + 1233 : 4 (1 + 333)

4 + 1233

Rewrite as

= 4 \cdot 1 + 4 \cdot 333

Factor out common term

4

= 4 (1 + 333)

= \frac{4 ( 1 + 3 33 )}{74}

Cancel the common factor:

2

= \frac{2 ( 1 + 3 33 )}{37}

The solutions to the quadratic equation are:

u = - \frac{2 ( 3 33 - 1 )}{37}, u = \frac{2 ( 1 + 3 33 )}{37}

Substitute back

u = cos (x)

cos (x) = - \frac{2 ( 3 33 - 1 )}{37}, cos (x) = \frac{2 ( 1 + 3 33 )}{37}

cos (x) = - \frac{2 ( 3 33 - 1 )}{37}, cos (x) = \frac{2 ( 1 + 3 33 )}{37}

cos (x) = - \frac{2 ( 3 33 - 1 )}{37} : x = arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn

cos (x) = - \frac{2 ( 3 33 - 1 )}{37}

Apply trig inverse properties

cos (x) = - \frac{2 ( 3 33 - 1 )}{37}

General solutions for

cos (x) = - \frac{2 ( 3 33 - 1 )}{37}

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

x = arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn

x = arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn

cos (x) = \frac{2 ( 1 + 3 33 )}{37} : x = arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn, x = 2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

cos (x) = \frac{2 ( 1 + 3 33 )}{37}

Apply trig inverse properties

cos (x) = \frac{2 ( 1 + 3 33 )}{37}

General solutions for

cos (x) = \frac{2 ( 1 + 3 33 )}{37}

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

x = arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn, x = 2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

x = arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn, x = 2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Combine all the solutions

x = arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn, x = - arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn, x = arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn, x = 2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn :

False

arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn

Plug in

n = 1

arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

For

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

plug in

x = arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

6 sin (arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) = cos (arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) - 2

Refine

2.87749 \dots = - 2.87749 \dots

\Rightarrow False

Check the solution

- arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn :

True

- arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn

Plug in

n = 1

- arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

For

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

plug in

x = - arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

6 sin (- arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) = cos (- arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) - 2

Refine

- 2.87749 \dots = - 2.87749 \dots

\Rightarrow True

Check the solution

arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn :

False

arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Plug in

n = 1

arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

For

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

plug in

x = arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

6 sin (arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) = cos (arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) - 2

Refine

1.01439 \dots = - 1.01439 \dots

\Rightarrow False

Check the solution

2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn :

True

2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

For

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

plug in

x = 2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

6 sin (2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) = cos (2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) - 2

Refine

- 1.01439 \dots = - 1.01439 \dots

\Rightarrow True

x = - arccos (- \frac{2 ( 3 33 - 1 )}{37}) + 2 πn, x = 2 π - arccos (\frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Show solutions in decimal form

x = - 2.64141 \dots + 2 πn, x = 2 π - 0.16988 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin(x)=cos(x)-2 ?
The general solution for 6sin(x)=cos(x)-2 is x=-2.64141…+2pin,x=2pi-0.16988…+2pin