What is the general solution for -6sin(x)-5cos(x)=2 ?

The general solution for -6sin(x)-5cos(x)=2 is x=2.70581…+2pin,x=2pi-0.95369…+2pin

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Popular Trigonometry >

-6sin(x)-5cos(x)=2

Solution

- 6 sin (x) - 5 cos (x) = 2

Solution

x = 2.70581 \dots + 2 πn, x = 2 π - 0.95369 \dots + 2 πn

Degrees

x = 155.03164 \dots^{\circ} + 36 0^{\circ} n, x = 305.35720 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 6 sin (x) - 5 cos (x) = 2

Add

5 cos (x)

to both sides

- 6 sin (x) = 2 + 5 cos (x)

Square both sides

(- 6 sin (x))^{2} = (2 + 5 cos (x))^{2}

Subtract

(2 + 5 cos (x))^{2}

from both sides

36 sin^{2} (x) - 4 - 20 cos (x) - 25 cos^{2} (x) = 0

Rewrite using trig identities

- 4 - 20 cos (x) - 25 cos^{2} (x) + 36 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 4 - 20 cos (x) - 25 cos^{2} (x) + 36 (1 - cos^{2} (x))

Simplify

- 4 - 20 cos (x) - 25 cos^{2} (x) + 36 (1 - cos^{2} (x)) : - 61 cos^{2} (x) - 20 cos (x) + 32

- 4 - 20 cos (x) - 25 cos^{2} (x) + 36 (1 - cos^{2} (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 - 20 cos (x) - 25 cos^{2} (x) + 36 - 36 cos^{2} (x)

Simplify

- 4 - 20 cos (x) - 25 cos^{2} (x) + 36 - 36 cos^{2} (x) : - 61 cos^{2} (x) - 20 cos (x) + 32

- 4 - 20 cos (x) - 25 cos^{2} (x) + 36 - 36 cos^{2} (x)

Group like terms

= - 20 cos (x) - 25 cos^{2} (x) - 36 cos^{2} (x) - 4 + 36

Add similar elements:

- 25 cos^{2} (x) - 36 cos^{2} (x) = - 61 cos^{2} (x)

= - 20 cos (x) - 61 cos^{2} (x) - 4 + 36

Add/Subtract the numbers:

- 4 + 36 = 32

= - 61 cos^{2} (x) - 20 cos (x) + 32

= - 61 cos^{2} (x) - 20 cos (x) + 32

= - 61 cos^{2} (x) - 20 cos (x) + 32

32 - 20 cos (x) - 61 cos^{2} (x) = 0

Solve by substitution

32 - 20 cos (x) - 61 cos^{2} (x) = 0

Let:

cos (x) = u

32 - 20 u - 61 u^{2} = 0

32 - 20 u - 61 u^{2} = 0 : u = - \frac{2 ( 5 + 3 57 )}{61}, u = \frac{2 ( 3 57 - 5 )}{61}

32 - 20 u - 61 u^{2} = 0

Write in the standard form

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

- 61 u^{2} - 20 u + 32 = 0

Solve with the quadratic formula

- 61 u^{2} - 20 u + 32 = 0

Quadratic Equation Formula:

For

a = - 61, b = - 20, c = 32

u_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 ( - 61 ) \cdot 32}{2 ( - 61 )}

u_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 ( - 61 ) \cdot 32}{2 ( - 61 )}

(- 20)^{2} - 4 (- 61) \cdot 32 = 1257

(- 20)^{2} - 4 (- 61) \cdot 32

Apply rule

- (- a) = a

= (- 20)^{2} + 4 \cdot 61 \cdot 32

Apply exponent rule:

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

n

is even

(- 20)^{2} = 2 0^{2}

= 2 0^{2} + 4 \cdot 61 \cdot 32

Multiply the numbers:

4 \cdot 61 \cdot 32 = 7808

= 2 0^{2} + 7808

2 0^{2} = 400

= 400 + 7808

Add the numbers:

400 + 7808 = 8208

= 8208

Prime factorization of

8208 : 2^{4} \cdot 3^{3} \cdot 19

8208

8208

divides by

28208 = 4104 \cdot 2

= 2 \cdot 4104

4104

divides by

24104 = 2052 \cdot 2

= 2 \cdot 2 \cdot 2052

2052

divides by

22052 = 1026 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 1026

1026

divides by

21026 = 513 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 513

513

divides by

3513 = 171 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 171

171

divides by

3171 = 57 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 57

57

divides by

357 = 19 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 19

2, 3, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 19

= 2^{4} \cdot 3^{3} \cdot 19

= 2^{4} \cdot 3^{3} \cdot 19

Apply exponent rule:

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

= 2^{4} \cdot 3^{2} \cdot 3 \cdot 19

Apply radical rule:

n ab = n a n b

= 2^{4} 3^{2} 3 \cdot 19

Apply radical rule:

n a^{m} = a^{\frac{m}{n}}

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

= 2^{2} 3^{2} 3 \cdot 19

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 2^{2} \cdot 3 3 \cdot 19

Refine

= 1257

u_{1, 2} = \frac{- ( - 20 ) \pm 12 57}{2 ( - 61 )}

Separate the solutions

u_{1} = \frac{- ( - 20 ) + 12 57}{2 ( - 61 )}, u_{2} = \frac{- ( - 20 ) - 12 57}{2 ( - 61 )}

u = \frac{- ( - 20 ) + 12 57}{2 ( - 61 )} : - \frac{2 ( 5 + 3 57 )}{61}

\frac{- ( - 20 ) + 12 57}{2 ( - 61 )}

Remove parentheses:

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

= \frac{20 + 12 57}{- 2 \cdot 61}

Multiply the numbers:

2 \cdot 61 = 122

= \frac{20 + 12 57}{- 122}

Apply the fraction rule:

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

= - \frac{20 + 12 57}{122}

Cancel

\frac{20 + 12 57}{122} : \frac{2 ( 5 + 3 57 )}{61}

\frac{20 + 12 57}{122}

Factor

20 + 1257 : 4 (5 + 357)

20 + 1257

Rewrite as

= 4 \cdot 5 + 4 \cdot 357

Factor out common term

4

= 4 (5 + 357)

= \frac{4 ( 5 + 3 57 )}{122}

Cancel the common factor:

2

= \frac{2 ( 5 + 3 57 )}{61}

= - \frac{2 ( 5 + 3 57 )}{61}

u = \frac{- ( - 20 ) - 12 57}{2 ( - 61 )} : \frac{2 ( 3 57 - 5 )}{61}

\frac{- ( - 20 ) - 12 57}{2 ( - 61 )}

Remove parentheses:

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

= \frac{20 - 12 57}{- 2 \cdot 61}

Multiply the numbers:

2 \cdot 61 = 122

= \frac{20 - 12 57}{- 122}

Apply the fraction rule:

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

20 - 1257 = - (1257 - 20)

= \frac{12 57 - 20}{122}

Factor

1257 - 20 : 4 (357 - 5)

1257 - 20

Rewrite as

= 4 \cdot 357 - 4 \cdot 5

Factor out common term

4

= 4 (357 - 5)

= \frac{4 ( 3 57 - 5 )}{122}

Cancel the common factor:

2

= \frac{2 ( 3 57 - 5 )}{61}

The solutions to the quadratic equation are:

u = - \frac{2 ( 5 + 3 57 )}{61}, u = \frac{2 ( 3 57 - 5 )}{61}

Substitute back

u = cos (x)

cos (x) = - \frac{2 ( 5 + 3 57 )}{61}, cos (x) = \frac{2 ( 3 57 - 5 )}{61}

cos (x) = - \frac{2 ( 5 + 3 57 )}{61}, cos (x) = \frac{2 ( 3 57 - 5 )}{61}

cos (x) = - \frac{2 ( 5 + 3 57 )}{61} : x = arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn, x = - arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn

cos (x) = - \frac{2 ( 5 + 3 57 )}{61}

Apply trig inverse properties

cos (x) = - \frac{2 ( 5 + 3 57 )}{61}

General solutions for

cos (x) = - \frac{2 ( 5 + 3 57 )}{61}

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

x = arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn, x = - arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn

x = arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn, x = - arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn

cos (x) = \frac{2 ( 3 57 - 5 )}{61} : x = arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn, x = 2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

cos (x) = \frac{2 ( 3 57 - 5 )}{61}

Apply trig inverse properties

cos (x) = \frac{2 ( 3 57 - 5 )}{61}

General solutions for

cos (x) = \frac{2 ( 3 57 - 5 )}{61}

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

x = arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn, x = 2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

x = arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn, x = 2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

Combine all the solutions

x = arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn, x = - arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn, x = arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn, x = 2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

- 6 sin (x) - 5 cos (x) = 2

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

Check the solution

arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn :

True

arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn

Plug in

n = 1

arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1

For

- 6 sin (x) - 5 cos (x) = 2

plug in

x = arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1

- 6 sin (arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1) - 5 cos (arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

Check the solution

- arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn :

False

- arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn

Plug in

n = 1

- arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1

For

- 6 sin (x) - 5 cos (x) = 2

plug in

x = - arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1

- 6 sin (- arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1) - 5 cos (- arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 π 1) = 2

Refine

7.06541 \dots = 2

\Rightarrow False

Check the solution

arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn :

False

arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

Plug in

n = 1

arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1

For

- 6 sin (x) - 5 cos (x) = 2

plug in

x = arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1

- 6 sin (arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1) - 5 cos (arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1) = 2

Refine

- 7.78672 \dots = 2

\Rightarrow False

Check the solution

2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn :

True

2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1

For

- 6 sin (x) - 5 cos (x) = 2

plug in

x = 2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1

- 6 sin (2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1) - 5 cos (2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

x = arccos (- \frac{2 ( 5 + 3 57 )}{61}) + 2 πn, x = 2 π - arccos (\frac{2 ( 3 57 - 5 )}{61}) + 2 πn

Show solutions in decimal form

x = 2.70581 \dots + 2 πn, x = 2 π - 0.95369 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -6sin(x)-5cos(x)=2 ?
The general solution for -6sin(x)-5cos(x)=2 is x=2.70581…+2pin,x=2pi-0.95369…+2pin