What is the general solution for 6sin(x)-2cos(x)=7 ?

The general solution for 6sin(x)-2cos(x)=7 is No Solution for x\in\mathbb{R}

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6sin(x)-2cos(x)=7

Solution

6 sin (x) - 2 cos (x) = 7

Solution

No Solution for x \in R

Solution steps

6 sin (x) - 2 cos (x) = 7

Add

2 cos (x)

to both sides

6 sin (x) = 7 + 2 cos (x)

Square both sides

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

Subtract

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

from both sides

36 sin^{2} (x) - 49 - 28 cos (x) - 4 cos^{2} (x) = 0

Rewrite using trig identities

- 49 - 28 cos (x) + 36 sin^{2} (x) - 4 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 49 - 28 cos (x) + 36 (1 - cos^{2} (x)) - 4 cos^{2} (x)

Simplify

- 49 - 28 cos (x) + 36 (1 - cos^{2} (x)) - 4 cos^{2} (x) : - 40 cos^{2} (x) - 28 cos (x) - 13

- 49 - 28 cos (x) + 36 (1 - cos^{2} (x)) - 4 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)

= - 49 - 28 cos (x) + 36 - 36 cos^{2} (x) - 4 cos^{2} (x)

Simplify

- 49 - 28 cos (x) + 36 - 36 cos^{2} (x) - 4 cos^{2} (x) : - 40 cos^{2} (x) - 28 cos (x) - 13

- 49 - 28 cos (x) + 36 - 36 cos^{2} (x) - 4 cos^{2} (x)

Add similar elements:

- 36 cos^{2} (x) - 4 cos^{2} (x) = - 40 cos^{2} (x)

= - 49 - 28 cos (x) + 36 - 40 cos^{2} (x)

Group like terms

= - 28 cos (x) - 40 cos^{2} (x) - 49 + 36

Add/Subtract the numbers:

- 49 + 36 = - 13

= - 40 cos^{2} (x) - 28 cos (x) - 13

= - 40 cos^{2} (x) - 28 cos (x) - 13

= - 40 cos^{2} (x) - 28 cos (x) - 13

- 13 - 28 cos (x) - 40 cos^{2} (x) = 0

Solve by substitution

- 13 - 28 cos (x) - 40 cos^{2} (x) = 0

Let:

cos (x) = u

- 13 - 28 u - 40 u^{2} = 0

- 13 - 28 u - 40 u^{2} = 0 : u = - \frac{7}{20} - i \frac{9}{20}, u = - \frac{7}{20} + i \frac{9}{20}

- 13 - 28 u - 40 u^{2} = 0

Write in the standard form

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

- 40 u^{2} - 28 u - 13 = 0

Solve with the quadratic formula

- 40 u^{2} - 28 u - 13 = 0

Quadratic Equation Formula:

For

a = - 40, b = - 28, c = - 13

u_{1, 2} = \frac{- ( - 28 ) \pm ( - 28 ) ^{2} - 4 ( - 40 ) ( - 13 )}{2 ( - 40 )}

u_{1, 2} = \frac{- ( - 28 ) \pm ( - 28 ) ^{2} - 4 ( - 40 ) ( - 13 )}{2 ( - 40 )}

Simplify

(- 28)^{2} - 4 (- 40) (- 13) : 36 i

(- 28)^{2} - 4 (- 40) (- 13)

Apply rule

- (- a) = a

= (- 28)^{2} - 4 \cdot 40 \cdot 13

Apply exponent rule:

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

n

is even

(- 28)^{2} = 2 8^{2}

= 2 8^{2} - 4 \cdot 40 \cdot 13

Multiply the numbers:

4 \cdot 40 \cdot 13 = 2080

= 2 8^{2} - 2080

Apply imaginary number rule:

- a = i a

= i 2080 - 2 8^{2}

- 2 8^{2} + 2080 = 36

- 2 8^{2} + 2080

2 8^{2} = 784

= - 784 + 2080

Add/Subtract the numbers:

- 784 + 2080 = 1296

= 1296

Factor the number:

1296 = 3 6^{2}

= 3 6^{2}

Apply radical rule:

n a^{n} = a

3 6^{2} = 36

= 36

= 36 i

u_{1, 2} = \frac{- ( - 28 ) \pm 36 i}{2 ( - 40 )}

Separate the solutions

u_{1} = \frac{- ( - 28 ) + 36 i}{2 ( - 40 )}, u_{2} = \frac{- ( - 28 ) - 36 i}{2 ( - 40 )}

u = \frac{- ( - 28 ) + 36 i}{2 ( - 40 )} : - \frac{7}{20} - i \frac{9}{20}

\frac{- ( - 28 ) + 36 i}{2 ( - 40 )}

Remove parentheses:

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

= \frac{28 + 36 i}{- 2 \cdot 40}

Multiply the numbers:

2 \cdot 40 = 80

= \frac{28 + 36 i}{- 80}

Apply the fraction rule:

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

= - \frac{28 + 36 i}{80}

Cancel

\frac{28 + 36 i}{80} : \frac{7 + 9 i}{20}

\frac{28 + 36 i}{80}

Factor

28 + 36 i : 4 (7 + 9 i)

28 + 36 i

Rewrite as

= 4 \cdot 7 + 4 \cdot 9 i

Factor out common term

4

= 4 (7 + 9 i)

= \frac{4 ( 7 + 9 i )}{80}

Cancel the common factor:

4

= \frac{7 + 9 i}{20}

= - \frac{7 + 9 i}{20}

Rewrite

- \frac{7 + 9 i}{20}

in standard complex form:

- \frac{7}{20} - \frac{9}{20} i

- \frac{7 + 9 i}{20}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{7 + 9 i}{20} = - (\frac{7}{20}) - (\frac{9 i}{20})

= - (\frac{7}{20}) - (\frac{9 i}{20})

Remove parentheses:

(a) = a

= - \frac{7}{20} - \frac{9 i}{20}

= - \frac{7}{20} - \frac{9}{20} i

u = \frac{- ( - 28 ) - 36 i}{2 ( - 40 )} : - \frac{7}{20} + i \frac{9}{20}

\frac{- ( - 28 ) - 36 i}{2 ( - 40 )}

Remove parentheses:

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

= \frac{28 - 36 i}{- 2 \cdot 40}

Multiply the numbers:

2 \cdot 40 = 80

= \frac{28 - 36 i}{- 80}

Apply the fraction rule:

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

= - \frac{28 - 36 i}{80}

Cancel

\frac{28 - 36 i}{80} : \frac{7 - 9 i}{20}

\frac{28 - 36 i}{80}

Factor

28 - 36 i : 4 (7 - 9 i)

28 - 36 i

Rewrite as

= 4 \cdot 7 - 4 \cdot 9 i

Factor out common term

4

= 4 (7 - 9 i)

= \frac{4 ( 7 - 9 i )}{80}

Cancel the common factor:

4

= \frac{7 - 9 i}{20}

= - \frac{7 - 9 i}{20}

Rewrite

- \frac{7 - 9 i}{20}

in standard complex form:

- \frac{7}{20} + \frac{9}{20} i

- \frac{7 - 9 i}{20}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{7 - 9 i}{20} = - (\frac{7}{20}) - (- \frac{9 i}{20})

= - (\frac{7}{20}) - (- \frac{9 i}{20})

Remove parentheses:

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

= - \frac{7}{20} + \frac{9 i}{20}

= - \frac{7}{20} + \frac{9}{20} i

The solutions to the quadratic equation are:

u = - \frac{7}{20} - i \frac{9}{20}, u = - \frac{7}{20} + i \frac{9}{20}

Substitute back

u = cos (x)

cos (x) = - \frac{7}{20} - i \frac{9}{20}, cos (x) = - \frac{7}{20} + i \frac{9}{20}

cos (x) = - \frac{7}{20} - i \frac{9}{20}, cos (x) = - \frac{7}{20} + i \frac{9}{20}

cos (x) = - \frac{7}{20} - i \frac{9}{20} :

No Solution

cos (x) = - \frac{7}{20} - i \frac{9}{20}

No Solution

cos (x) = - \frac{7}{20} + i \frac{9}{20} :

No Solution

cos (x) = - \frac{7}{20} + i \frac{9}{20}

No Solution

Combine all the solutions

No Solution

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

6 sin (x) - 2 cos (x) = 7

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

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin(x)-2cos(x)=7 ?
The general solution for 6sin(x)-2cos(x)=7 is No Solution for x\in\mathbb{R}