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

The general solution for 3sin(x)+6cos(x)=4 is x=1.39557…+2pin,x=2pi-0.46828…+2pin

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Popular Trigonometry >

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

Solution

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

Solution

x = 1.39557 \dots + 2 πn, x = 2 π - 0.46828 \dots + 2 πn

Degrees

x = 79.96077 \dots^{\circ} + 36 0^{\circ} n, x = 333.16932 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

6 cos (x)

from both sides

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

Square both sides

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

Subtract

(4 - 6 cos (x))^{2}

from both sides

9 sin^{2} (x) - 16 + 48 cos (x) - 36 cos^{2} (x) = 0

Rewrite using trig identities

- 16 - 36 cos^{2} (x) + 48 cos (x) + 9 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 16 - 36 cos^{2} (x) + 48 cos (x) + 9 (1 - cos^{2} (x))

Simplify

- 16 - 36 cos^{2} (x) + 48 cos (x) + 9 (1 - cos^{2} (x)) : 48 cos (x) - 45 cos^{2} (x) - 7

- 16 - 36 cos^{2} (x) + 48 cos (x) + 9 (1 - cos^{2} (x))

Expand

9 (1 - cos^{2} (x)) : 9 - 9 cos^{2} (x)

9 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = cos^{2} (x)

= 9 \cdot 1 - 9 cos^{2} (x)

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 cos^{2} (x)

= - 16 - 36 cos^{2} (x) + 48 cos (x) + 9 - 9 cos^{2} (x)

Simplify

- 16 - 36 cos^{2} (x) + 48 cos (x) + 9 - 9 cos^{2} (x) : 48 cos (x) - 45 cos^{2} (x) - 7

- 16 - 36 cos^{2} (x) + 48 cos (x) + 9 - 9 cos^{2} (x)

Group like terms

= - 36 cos^{2} (x) + 48 cos (x) - 9 cos^{2} (x) - 16 + 9

Add similar elements:

- 36 cos^{2} (x) - 9 cos^{2} (x) = - 45 cos^{2} (x)

= - 45 cos^{2} (x) + 48 cos (x) - 16 + 9

Add/Subtract the numbers:

- 16 + 9 = - 7

= 48 cos (x) - 45 cos^{2} (x) - 7

= 48 cos (x) - 45 cos^{2} (x) - 7

= 48 cos (x) - 45 cos^{2} (x) - 7

- 7 - 45 cos^{2} (x) + 48 cos (x) = 0

Solve by substitution

- 7 - 45 cos^{2} (x) + 48 cos (x) = 0

Let:

cos (x) = u

- 7 - 45 u^{2} + 48 u = 0

- 7 - 45 u^{2} + 48 u = 0 : u = \frac{8 - 29}{15}, u = \frac{8 + 29}{15}

- 7 - 45 u^{2} + 48 u = 0

Write in the standard form

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

- 45 u^{2} + 48 u - 7 = 0

Solve with the quadratic formula

- 45 u^{2} + 48 u - 7 = 0

Quadratic Equation Formula:

For

a = - 45, b = 48, c = - 7

u_{1, 2} = \frac{- 48 \pm 4 8 ^{2} - 4 ( - 45 ) ( - 7 )}{2 ( - 45 )}

u_{1, 2} = \frac{- 48 \pm 4 8 ^{2} - 4 ( - 45 ) ( - 7 )}{2 ( - 45 )}

4 8^{2} - 4 (- 45) (- 7) = 629

4 8^{2} - 4 (- 45) (- 7)

Apply rule

- (- a) = a

= 4 8^{2} - 4 \cdot 45 \cdot 7

Multiply the numbers:

4 \cdot 45 \cdot 7 = 1260

= 4 8^{2} - 1260

4 8^{2} = 2304

= 2304 - 1260

Subtract the numbers:

2304 - 1260 = 1044

= 1044

Prime factorization of

1044 : 2^{2} \cdot 3^{2} \cdot 29

1044

1044

divides by

21044 = 522 \cdot 2

= 2 \cdot 522

522

divides by

2522 = 261 \cdot 2

= 2 \cdot 2 \cdot 261

261

divides by

3261 = 87 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 87

87

divides by

387 = 29 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 29

2, 3, 29

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 29

= 2^{2} \cdot 3^{2} \cdot 29

= 2^{2} \cdot 3^{2} \cdot 29

Apply radical rule:

n ab = n a n b

= 29 2^{2} 3^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 229 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 2 \cdot 329

Refine

= 629

u_{1, 2} = \frac{- 48 \pm 6 29}{2 ( - 45 )}

Separate the solutions

u_{1} = \frac{- 48 + 6 29}{2 ( - 45 )}, u_{2} = \frac{- 48 - 6 29}{2 ( - 45 )}

u = \frac{- 48 + 6 29}{2 ( - 45 )} : \frac{8 - 29}{15}

\frac{- 48 + 6 29}{2 ( - 45 )}

Remove parentheses:

(- a) = - a

= \frac{- 48 + 6 29}{- 2 \cdot 45}

Multiply the numbers:

2 \cdot 45 = 90

= \frac{- 48 + 6 29}{- 90}

Apply the fraction rule:

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

- 48 + 629 = - (48 - 629)

= \frac{48 - 6 29}{90}

Factor

48 - 629 : 6 (8 - 29)

48 - 629

Rewrite as

= 6 \cdot 8 - 629

Factor out common term

6

= 6 (8 - 29)

= \frac{6 ( 8 - 29 )}{90}

Cancel the common factor:

6

= \frac{8 - 29}{15}

u = \frac{- 48 - 6 29}{2 ( - 45 )} : \frac{8 + 29}{15}

\frac{- 48 - 6 29}{2 ( - 45 )}

Remove parentheses:

(- a) = - a

= \frac{- 48 - 6 29}{- 2 \cdot 45}

Multiply the numbers:

2 \cdot 45 = 90

= \frac{- 48 - 6 29}{- 90}

Apply the fraction rule:

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

- 48 - 629 = - (48 + 629)

= \frac{48 + 6 29}{90}

Factor

48 + 629 : 6 (8 + 29)

48 + 629

Rewrite as

= 6 \cdot 8 + 629

Factor out common term

6

= 6 (8 + 29)

= \frac{6 ( 8 + 29 )}{90}

Cancel the common factor:

6

= \frac{8 + 29}{15}

The solutions to the quadratic equation are:

u = \frac{8 - 29}{15}, u = \frac{8 + 29}{15}

Substitute back

u = cos (x)

cos (x) = \frac{8 - 29}{15}, cos (x) = \frac{8 + 29}{15}

cos (x) = \frac{8 - 29}{15}, cos (x) = \frac{8 + 29}{15}

cos (x) = \frac{8 - 29}{15} : x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 - 29}{15}) + 2 πn

cos (x) = \frac{8 - 29}{15}

Apply trig inverse properties

cos (x) = \frac{8 - 29}{15}

General solutions for

cos (x) = \frac{8 - 29}{15}

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

x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 - 29}{15}) + 2 πn

x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 - 29}{15}) + 2 πn

cos (x) = \frac{8 + 29}{15} : x = arccos (\frac{8 + 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

cos (x) = \frac{8 + 29}{15}

Apply trig inverse properties

cos (x) = \frac{8 + 29}{15}

General solutions for

cos (x) = \frac{8 + 29}{15}

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

x = arccos (\frac{8 + 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

x = arccos (\frac{8 + 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Combine all the solutions

x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 - 29}{15}) + 2 πn, x = arccos (\frac{8 + 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

arccos (\frac{8 - 29}{15}) + 2 πn :

True

arccos (\frac{8 - 29}{15}) + 2 πn

Plug in

n = 1

arccos (\frac{8 - 29}{15}) + 2 π 1

For

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

plug in

x = arccos (\frac{8 - 29}{15}) + 2 π 1

3 sin (arccos (\frac{8 - 29}{15}) + 2 π 1) + 6 cos (arccos (\frac{8 - 29}{15}) + 2 π 1) = 4

Refine

4 = 4

\Rightarrow True

Check the solution

2 π - arccos (\frac{8 - 29}{15}) + 2 πn :

False

2 π - arccos (\frac{8 - 29}{15}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{8 - 29}{15}) + 2 π 1

For

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

plug in

x = 2 π - arccos (\frac{8 - 29}{15}) + 2 π 1

3 sin (2 π - arccos (\frac{8 - 29}{15}) + 2 π 1) + 6 cos (2 π - arccos (\frac{8 - 29}{15}) + 2 π 1) = 4

Refine

- 1.90813 \dots = 4

\Rightarrow False

Check the solution

arccos (\frac{8 + 29}{15}) + 2 πn :

False

arccos (\frac{8 + 29}{15}) + 2 πn

Plug in

n = 1

arccos (\frac{8 + 29}{15}) + 2 π 1

For

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

plug in

x = arccos (\frac{8 + 29}{15}) + 2 π 1

3 sin (arccos (\frac{8 + 29}{15}) + 2 π 1) + 6 cos (arccos (\frac{8 + 29}{15}) + 2 π 1) = 4

Refine

6.70813 \dots = 4

\Rightarrow False

Check the solution

2 π - arccos (\frac{8 + 29}{15}) + 2 πn :

True

2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{8 + 29}{15}) + 2 π 1

For

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

plug in

x = 2 π - arccos (\frac{8 + 29}{15}) + 2 π 1

3 sin (2 π - arccos (\frac{8 + 29}{15}) + 2 π 1) + 6 cos (2 π - arccos (\frac{8 + 29}{15}) + 2 π 1) = 4

Refine

4 = 4

\Rightarrow True

x = arccos (\frac{8 - 29}{15}) + 2 πn, x = 2 π - arccos (\frac{8 + 29}{15}) + 2 πn

Show solutions in decimal form

x = 1.39557 \dots + 2 πn, x = 2 π - 0.46828 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(x)+6cos(x)=4 ?
The general solution for 3sin(x)+6cos(x)=4 is x=1.39557…+2pin,x=2pi-0.46828…+2pin