What is the general solution for 4sin(x)-6cos(x)=3 ?

The general solution for 4sin(x)-6cos(x)=3 is x=-2.58786…+2pin,x=1.41186…+2pin

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Popular Trigonometry >

4sin(x)-6cos(x)=3

Solution

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

Solution

x = - 2.58786 \dots + 2 πn, x = 1.41186 \dots + 2 πn

Degrees

x = - 148.27395 \dots^{\circ} + 36 0^{\circ} n, x = 80.89382 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Add

6 cos (x)

to both sides

4 sin (x) = 3 + 6 cos (x)

Square both sides

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

Subtract

(3 + 6 cos (x))^{2}

from both sides

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

Rewrite using trig identities

- 9 + 16 sin^{2} (x) - 36 cos (x) - 36 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 9 + 16 (1 - cos^{2} (x)) - 36 cos (x) - 36 cos^{2} (x)

Simplify

- 9 + 16 (1 - cos^{2} (x)) - 36 cos (x) - 36 cos^{2} (x) : - 52 cos^{2} (x) - 36 cos (x) + 7

- 9 + 16 (1 - cos^{2} (x)) - 36 cos (x) - 36 cos^{2} (x)

Expand

16 (1 - cos^{2} (x)) : 16 - 16 cos^{2} (x)

16 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 16, b = 1, c = cos^{2} (x)

= 16 \cdot 1 - 16 cos^{2} (x)

Multiply the numbers:

16 \cdot 1 = 16

= 16 - 16 cos^{2} (x)

= - 9 + 16 - 16 cos^{2} (x) - 36 cos (x) - 36 cos^{2} (x)

Simplify

- 9 + 16 - 16 cos^{2} (x) - 36 cos (x) - 36 cos^{2} (x) : - 52 cos^{2} (x) - 36 cos (x) + 7

- 9 + 16 - 16 cos^{2} (x) - 36 cos (x) - 36 cos^{2} (x)

Group like terms

= - 16 cos^{2} (x) - 36 cos (x) - 36 cos^{2} (x) - 9 + 16

Add similar elements:

- 16 cos^{2} (x) - 36 cos^{2} (x) = - 52 cos^{2} (x)

= - 52 cos^{2} (x) - 36 cos (x) - 9 + 16

Add/Subtract the numbers:

- 9 + 16 = 7

= - 52 cos^{2} (x) - 36 cos (x) + 7

= - 52 cos^{2} (x) - 36 cos (x) + 7

= - 52 cos^{2} (x) - 36 cos (x) + 7

7 - 36 cos (x) - 52 cos^{2} (x) = 0

Solve by substitution

7 - 36 cos (x) - 52 cos^{2} (x) = 0

Let:

cos (x) = u

7 - 36 u - 52 u^{2} = 0

7 - 36 u - 52 u^{2} = 0 : u = - \frac{9 + 2 43}{26}, u = \frac{2 43 - 9}{26}

7 - 36 u - 52 u^{2} = 0

Write in the standard form

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

- 52 u^{2} - 36 u + 7 = 0

Solve with the quadratic formula

- 52 u^{2} - 36 u + 7 = 0

Quadratic Equation Formula:

For

a = - 52, b = - 36, c = 7

u_{1, 2} = \frac{- ( - 36 ) \pm ( - 36 ) ^{2} - 4 ( - 52 ) \cdot 7}{2 ( - 52 )}

u_{1, 2} = \frac{- ( - 36 ) \pm ( - 36 ) ^{2} - 4 ( - 52 ) \cdot 7}{2 ( - 52 )}

(- 36)^{2} - 4 (- 52) \cdot 7 = 843

(- 36)^{2} - 4 (- 52) \cdot 7

Apply rule

- (- a) = a

= (- 36)^{2} + 4 \cdot 52 \cdot 7

Apply exponent rule:

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

n

is even

(- 36)^{2} = 3 6^{2}

= 3 6^{2} + 4 \cdot 52 \cdot 7

Multiply the numbers:

4 \cdot 52 \cdot 7 = 1456

= 3 6^{2} + 1456

3 6^{2} = 1296

= 1296 + 1456

Add the numbers:

1296 + 1456 = 2752

= 2752

Prime factorization of

2752 : 2^{6} \cdot 43

2752

2752

divides by

22752 = 1376 \cdot 2

= 2 \cdot 1376

1376

divides by

21376 = 688 \cdot 2

= 2 \cdot 2 \cdot 688

688

divides by

2688 = 344 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 344

344

divides by

2344 = 172 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 172

172

divides by

2172 = 86 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 86

86

divides by

286 = 43 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 43

2, 43

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 43

= 2^{6} \cdot 43

= 2^{6} \cdot 43

Apply radical rule:

n ab = n a n b

= 43 2^{6}

Apply radical rule:

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

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

= 2^{3} 43

Refine

= 843

u_{1, 2} = \frac{- ( - 36 ) \pm 8 43}{2 ( - 52 )}

Separate the solutions

u_{1} = \frac{- ( - 36 ) + 8 43}{2 ( - 52 )}, u_{2} = \frac{- ( - 36 ) - 8 43}{2 ( - 52 )}

u = \frac{- ( - 36 ) + 8 43}{2 ( - 52 )} : - \frac{9 + 2 43}{26}

\frac{- ( - 36 ) + 8 43}{2 ( - 52 )}

Remove parentheses:

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

= \frac{36 + 8 43}{- 2 \cdot 52}

Multiply the numbers:

2 \cdot 52 = 104

= \frac{36 + 8 43}{- 104}

Apply the fraction rule:

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

= - \frac{36 + 8 43}{104}

Cancel

\frac{36 + 8 43}{104} : \frac{9 + 2 43}{26}

\frac{36 + 8 43}{104}

Factor

36 + 843 : 4 (9 + 243)

36 + 843

Rewrite as

= 4 \cdot 9 + 4 \cdot 243

Factor out common term

4

= 4 (9 + 243)

= \frac{4 ( 9 + 2 43 )}{104}

Cancel the common factor:

4

= \frac{9 + 2 43}{26}

= - \frac{9 + 2 43}{26}

u = \frac{- ( - 36 ) - 8 43}{2 ( - 52 )} : \frac{2 43 - 9}{26}

\frac{- ( - 36 ) - 8 43}{2 ( - 52 )}

Remove parentheses:

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

= \frac{36 - 8 43}{- 2 \cdot 52}

Multiply the numbers:

2 \cdot 52 = 104

= \frac{36 - 8 43}{- 104}

Apply the fraction rule:

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

36 - 843 = - (843 - 36)

= \frac{8 43 - 36}{104}

Factor

843 - 36 : 4 (243 - 9)

843 - 36

Rewrite as

= 4 \cdot 243 - 4 \cdot 9

Factor out common term

4

= 4 (243 - 9)

= \frac{4 ( 2 43 - 9 )}{104}

Cancel the common factor:

4

= \frac{2 43 - 9}{26}

The solutions to the quadratic equation are:

u = - \frac{9 + 2 43}{26}, u = \frac{2 43 - 9}{26}

Substitute back

u = cos (x)

cos (x) = - \frac{9 + 2 43}{26}, cos (x) = \frac{2 43 - 9}{26}

cos (x) = - \frac{9 + 2 43}{26}, cos (x) = \frac{2 43 - 9}{26}

cos (x) = - \frac{9 + 2 43}{26} : x = arccos (- \frac{9 + 2 43}{26}) + 2 πn, x = - arccos (- \frac{9 + 2 43}{26}) + 2 πn

cos (x) = - \frac{9 + 2 43}{26}

Apply trig inverse properties

cos (x) = - \frac{9 + 2 43}{26}

General solutions for

cos (x) = - \frac{9 + 2 43}{26}

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

x = arccos (- \frac{9 + 2 43}{26}) + 2 πn, x = - arccos (- \frac{9 + 2 43}{26}) + 2 πn

x = arccos (- \frac{9 + 2 43}{26}) + 2 πn, x = - arccos (- \frac{9 + 2 43}{26}) + 2 πn

cos (x) = \frac{2 43 - 9}{26} : x = arccos (\frac{2 43 - 9}{26}) + 2 πn, x = 2 π - arccos (\frac{2 43 - 9}{26}) + 2 πn

cos (x) = \frac{2 43 - 9}{26}

Apply trig inverse properties

cos (x) = \frac{2 43 - 9}{26}

General solutions for

cos (x) = \frac{2 43 - 9}{26}

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

x = arccos (\frac{2 43 - 9}{26}) + 2 πn, x = 2 π - arccos (\frac{2 43 - 9}{26}) + 2 πn

x = arccos (\frac{2 43 - 9}{26}) + 2 πn, x = 2 π - arccos (\frac{2 43 - 9}{26}) + 2 πn

Combine all the solutions

x = arccos (- \frac{9 + 2 43}{26}) + 2 πn, x = - arccos (- \frac{9 + 2 43}{26}) + 2 πn, x = arccos (\frac{2 43 - 9}{26}) + 2 πn, x = 2 π - arccos (\frac{2 43 - 9}{26}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

arccos (- \frac{9 + 2 43}{26}) + 2 πn :

False

arccos (- \frac{9 + 2 43}{26}) + 2 πn

Plug in

n = 1

arccos (- \frac{9 + 2 43}{26}) + 2 π 1

For

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

plug in

x = arccos (- \frac{9 + 2 43}{26}) + 2 π 1

4 sin (arccos (- \frac{9 + 2 43}{26}) + 2 π 1) - 6 cos (arccos (- \frac{9 + 2 43}{26}) + 2 π 1) = 3

Refine

7.20686 \dots = 3

\Rightarrow False

Check the solution

- arccos (- \frac{9 + 2 43}{26}) + 2 πn :

True

- arccos (- \frac{9 + 2 43}{26}) + 2 πn

Plug in

n = 1

- arccos (- \frac{9 + 2 43}{26}) + 2 π 1

For

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

plug in

x = - arccos (- \frac{9 + 2 43}{26}) + 2 π 1

4 sin (- arccos (- \frac{9 + 2 43}{26}) + 2 π 1) - 6 cos (- arccos (- \frac{9 + 2 43}{26}) + 2 π 1) = 3

Refine

3 = 3

\Rightarrow True

Check the solution

arccos (\frac{2 43 - 9}{26}) + 2 πn :

True

arccos (\frac{2 43 - 9}{26}) + 2 πn

Plug in

n = 1

arccos (\frac{2 43 - 9}{26}) + 2 π 1

For

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

plug in

x = arccos (\frac{2 43 - 9}{26}) + 2 π 1

4 sin (arccos (\frac{2 43 - 9}{26}) + 2 π 1) - 6 cos (arccos (\frac{2 43 - 9}{26}) + 2 π 1) = 3

Refine

3 = 3

\Rightarrow True

Check the solution

2 π - arccos (\frac{2 43 - 9}{26}) + 2 πn :

False

2 π - arccos (\frac{2 43 - 9}{26}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{2 43 - 9}{26}) + 2 π 1

For

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

plug in

x = 2 π - arccos (\frac{2 43 - 9}{26}) + 2 π 1

4 sin (2 π - arccos (\frac{2 43 - 9}{26}) + 2 π 1) - 6 cos (2 π - arccos (\frac{2 43 - 9}{26}) + 2 π 1) = 3

Refine

- 4.89917 \dots = 3

\Rightarrow False

x = - arccos (- \frac{9 + 2 43}{26}) + 2 πn, x = arccos (\frac{2 43 - 9}{26}) + 2 πn

Show solutions in decimal form

x = - 2.58786 \dots + 2 πn, x = 1.41186 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin(x)-6cos(x)=3 ?
The general solution for 4sin(x)-6cos(x)=3 is x=-2.58786…+2pin,x=1.41186…+2pin