What is the general solution for 6sin(x)-5cos(x)=7 ?

The general solution for 6sin(x)-5cos(x)=7 is x=2.72507…+2pin,x=1.80599…+2pin

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Popular Trigonometry >

6sin(x)-5cos(x)=7

Solution

6 sin (x) - 5 cos (x) = 7

Solution

x = 2.72507 \dots + 2 πn, x = 1.80599 \dots + 2 πn

Degrees

x = 156.13507 \dots^{\circ} + 36 0^{\circ} n, x = 103.47606 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 sin (x) - 5 cos (x) = 7

Add

5 cos (x)

to both sides

6 sin (x) = 7 + 5 cos (x)

Square both sides

(6 sin (x))^{2} = (7 + 5 cos (x))^{2}

Subtract

(7 + 5 cos (x))^{2}

from both sides

36 sin^{2} (x) - 49 - 70 cos (x) - 25 cos^{2} (x) = 0

Rewrite using trig identities

- 49 - 25 cos^{2} (x) + 36 sin^{2} (x) - 70 cos (x)

Use the Pythagorean identity:

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

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

= - 49 - 25 cos^{2} (x) + 36 (1 - cos^{2} (x)) - 70 cos (x)

Simplify

- 49 - 25 cos^{2} (x) + 36 (1 - cos^{2} (x)) - 70 cos (x) : - 61 cos^{2} (x) - 70 cos (x) - 13

- 49 - 25 cos^{2} (x) + 36 (1 - cos^{2} (x)) - 70 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)

= - 49 - 25 cos^{2} (x) + 36 - 36 cos^{2} (x) - 70 cos (x)

Simplify

- 49 - 25 cos^{2} (x) + 36 - 36 cos^{2} (x) - 70 cos (x) : - 61 cos^{2} (x) - 70 cos (x) - 13

- 49 - 25 cos^{2} (x) + 36 - 36 cos^{2} (x) - 70 cos (x)

Group like terms

= - 25 cos^{2} (x) - 36 cos^{2} (x) - 70 cos (x) - 49 + 36

Add similar elements:

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

= - 61 cos^{2} (x) - 70 cos (x) - 49 + 36

Add/Subtract the numbers:

- 49 + 36 = - 13

= - 61 cos^{2} (x) - 70 cos (x) - 13

= - 61 cos^{2} (x) - 70 cos (x) - 13

= - 61 cos^{2} (x) - 70 cos (x) - 13

- 13 - 61 cos^{2} (x) - 70 cos (x) = 0

Solve by substitution

- 13 - 61 cos^{2} (x) - 70 cos (x) = 0

Let:

cos (x) = u

- 13 - 61 u^{2} - 70 u = 0

- 13 - 61 u^{2} - 70 u = 0 : u = - \frac{35 + 12 3}{61}, u = - \frac{35 - 12 3}{61}

- 13 - 61 u^{2} - 70 u = 0

Write in the standard form

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

- 61 u^{2} - 70 u - 13 = 0

Solve with the quadratic formula

- 61 u^{2} - 70 u - 13 = 0

Quadratic Equation Formula:

For

a = - 61, b = - 70, c = - 13

u_{1, 2} = \frac{- ( - 70 ) \pm ( - 70 ) ^{2} - 4 ( - 61 ) ( - 13 )}{2 ( - 61 )}

u_{1, 2} = \frac{- ( - 70 ) \pm ( - 70 ) ^{2} - 4 ( - 61 ) ( - 13 )}{2 ( - 61 )}

(- 70)^{2} - 4 (- 61) (- 13) = 243

(- 70)^{2} - 4 (- 61) (- 13)

Apply rule

- (- a) = a

= (- 70)^{2} - 4 \cdot 61 \cdot 13

Apply exponent rule:

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

n

is even

(- 70)^{2} = 7 0^{2}

= 7 0^{2} - 4 \cdot 61 \cdot 13

Multiply the numbers:

4 \cdot 61 \cdot 13 = 3172

= 7 0^{2} - 3172

7 0^{2} = 4900

= 4900 - 3172

Subtract the numbers:

4900 - 3172 = 1728

= 1728

Prime factorization of

1728 : 2^{6} \cdot 3^{3}

1728

1728

divides by

21728 = 864 \cdot 2

= 2 \cdot 864

864

divides by

2864 = 432 \cdot 2

= 2 \cdot 2 \cdot 432

432

divides by

2432 = 216 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 216

216

divides by

2216 = 108 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 108

108

divides by

2108 = 54 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 54

54

divides by

254 = 27 \cdot 2

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

27

divides by

327 = 9 \cdot 3

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

9

divides by

39 = 3 \cdot 3

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

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

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

Apply exponent rule:

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

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

Apply radical rule:

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

Apply radical rule:

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

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

Apply radical rule:

3^{2} = 3

= 2^{3} \cdot 33

Refine

= 243

u_{1, 2} = \frac{- ( - 70 ) \pm 24 3}{2 ( - 61 )}

Separate the solutions

u_{1} = \frac{- ( - 70 ) + 24 3}{2 ( - 61 )}, u_{2} = \frac{- ( - 70 ) - 24 3}{2 ( - 61 )}

u = \frac{- ( - 70 ) + 24 3}{2 ( - 61 )} : - \frac{35 + 12 3}{61}

\frac{- ( - 70 ) + 24 3}{2 ( - 61 )}

Remove parentheses:

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

= \frac{70 + 24 3}{- 2 \cdot 61}

Multiply the numbers:

2 \cdot 61 = 122

= \frac{70 + 24 3}{- 122}

Apply the fraction rule:

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

= - \frac{70 + 24 3}{122}

Cancel

\frac{70 + 24 3}{122} : \frac{35 + 12 3}{61}

\frac{70 + 24 3}{122}

Factor

70 + 243 : 2 (35 + 123)

70 + 243

Rewrite as

= 2 \cdot 35 + 2 \cdot 123

Factor out common term

2

= 2 (35 + 123)

= \frac{2 ( 35 + 12 3 )}{122}

Cancel the common factor:

2

= \frac{35 + 12 3}{61}

= - \frac{35 + 12 3}{61}

u = \frac{- ( - 70 ) - 24 3}{2 ( - 61 )} : - \frac{35 - 12 3}{61}

\frac{- ( - 70 ) - 24 3}{2 ( - 61 )}

Remove parentheses:

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

= \frac{70 - 24 3}{- 2 \cdot 61}

Multiply the numbers:

2 \cdot 61 = 122

= \frac{70 - 24 3}{- 122}

Apply the fraction rule:

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

= - \frac{70 - 24 3}{122}

Cancel

\frac{70 - 24 3}{122} : \frac{35 - 12 3}{61}

\frac{70 - 24 3}{122}

Factor

70 - 243 : 2 (35 - 123)

70 - 243

Rewrite as

= 2 \cdot 35 - 2 \cdot 123

Factor out common term

2

= 2 (35 - 123)

= \frac{2 ( 35 - 12 3 )}{122}

Cancel the common factor:

2

= \frac{35 - 12 3}{61}

= - \frac{35 - 12 3}{61}

The solutions to the quadratic equation are:

u = - \frac{35 + 12 3}{61}, u = - \frac{35 - 12 3}{61}

Substitute back

u = cos (x)

cos (x) = - \frac{35 + 12 3}{61}, cos (x) = - \frac{35 - 12 3}{61}

cos (x) = - \frac{35 + 12 3}{61}, cos (x) = - \frac{35 - 12 3}{61}

cos (x) = - \frac{35 + 12 3}{61} : x = arccos (- \frac{35 + 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 + 12 3}{61}) + 2 πn

cos (x) = - \frac{35 + 12 3}{61}

Apply trig inverse properties

cos (x) = - \frac{35 + 12 3}{61}

General solutions for

cos (x) = - \frac{35 + 12 3}{61}

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

x = arccos (- \frac{35 + 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 + 12 3}{61}) + 2 πn

x = arccos (- \frac{35 + 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 + 12 3}{61}) + 2 πn

cos (x) = - \frac{35 - 12 3}{61} : x = arccos (- \frac{35 - 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 - 12 3}{61}) + 2 πn

cos (x) = - \frac{35 - 12 3}{61}

Apply trig inverse properties

cos (x) = - \frac{35 - 12 3}{61}

General solutions for

cos (x) = - \frac{35 - 12 3}{61}

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

x = arccos (- \frac{35 - 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 - 12 3}{61}) + 2 πn

x = arccos (- \frac{35 - 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 - 12 3}{61}) + 2 πn

Combine all the solutions

x = arccos (- \frac{35 + 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 + 12 3}{61}) + 2 πn, x = arccos (- \frac{35 - 12 3}{61}) + 2 πn, x = - arccos (- \frac{35 - 12 3}{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) = 7

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

Check the solution

arccos (- \frac{35 + 12 3}{61}) + 2 πn :

True

arccos (- \frac{35 + 12 3}{61}) + 2 πn

Plug in

n = 1

arccos (- \frac{35 + 12 3}{61}) + 2 π 1

For

6 sin (x) - 5 cos (x) = 7

plug in

x = arccos (- \frac{35 + 12 3}{61}) + 2 π 1

6 sin (arccos (- \frac{35 + 12 3}{61}) + 2 π 1) - 5 cos (arccos (- \frac{35 + 12 3}{61}) + 2 π 1) = 7

Refine

7 = 7

\Rightarrow True

Check the solution

- arccos (- \frac{35 + 12 3}{61}) + 2 πn :

False

- arccos (- \frac{35 + 12 3}{61}) + 2 πn

Plug in

n = 1

- arccos (- \frac{35 + 12 3}{61}) + 2 π 1

For

6 sin (x) - 5 cos (x) = 7

plug in

x = - arccos (- \frac{35 + 12 3}{61}) + 2 π 1

6 sin (- arccos (- \frac{35 + 12 3}{61}) + 2 π 1) - 5 cos (- arccos (- \frac{35 + 12 3}{61}) + 2 π 1) = 7

Refine

2.14501 \dots = 7

\Rightarrow False

Check the solution

arccos (- \frac{35 - 12 3}{61}) + 2 πn :

True

arccos (- \frac{35 - 12 3}{61}) + 2 πn

Plug in

n = 1

arccos (- \frac{35 - 12 3}{61}) + 2 π 1

For

6 sin (x) - 5 cos (x) = 7

plug in

x = arccos (- \frac{35 - 12 3}{61}) + 2 π 1

6 sin (arccos (- \frac{35 - 12 3}{61}) + 2 π 1) - 5 cos (arccos (- \frac{35 - 12 3}{61}) + 2 π 1) = 7

Refine

7 = 7

\Rightarrow True

Check the solution

- arccos (- \frac{35 - 12 3}{61}) + 2 πn :

False

- arccos (- \frac{35 - 12 3}{61}) + 2 πn

Plug in

n = 1

- arccos (- \frac{35 - 12 3}{61}) + 2 π 1

For

6 sin (x) - 5 cos (x) = 7

plug in

x = - arccos (- \frac{35 - 12 3}{61}) + 2 π 1

6 sin (- arccos (- \frac{35 - 12 3}{61}) + 2 π 1) - 5 cos (- arccos (- \frac{35 - 12 3}{61}) + 2 π 1) = 7

Refine

- 4.66960 \dots = 7

\Rightarrow False

x = arccos (- \frac{35 + 12 3}{61}) + 2 πn, x = arccos (- \frac{35 - 12 3}{61}) + 2 πn

Show solutions in decimal form

x = 2.72507 \dots + 2 πn, x = 1.80599 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin(x)-5cos(x)=7 ?
The general solution for 6sin(x)-5cos(x)=7 is x=2.72507…+2pin,x=1.80599…+2pin