What is the general solution for 3*sin(x)+4*cos(x)=2 ?

The general solution for 3*sin(x)+4*cos(x)=2 is x=1.80278…+2pin,x=2pi-0.51577…+2pin

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3sin(x)+4cos(x)=2

Solution

3 \cdot sin (x) + 4 \cdot cos (x) = 2

Solution

x = 1.80278 \dots + 2 πn, x = 2 π - 0.51577 \dots + 2 πn

Degrees

x = 103.29171 \dots^{\circ} + 36 0^{\circ} n, x = 330.44807 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (x) + 4 cos (x) = 2

Subtract

4 cos (x)

from both sides

3 sin (x) = 2 - 4 cos (x)

Square both sides

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

Subtract

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

from both sides

9 sin^{2} (x) - 4 + 16 cos (x) - 16 cos^{2} (x) = 0

Rewrite using trig identities

- 4 + 16 cos (x) - 16 cos^{2} (x) + 9 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 4 + 16 cos (x) - 16 cos^{2} (x) + 9 (1 - cos^{2} (x))

Simplify

- 4 + 16 cos (x) - 16 cos^{2} (x) + 9 (1 - cos^{2} (x)) : 16 cos (x) - 25 cos^{2} (x) + 5

- 4 + 16 cos (x) - 16 cos^{2} (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)

= - 4 + 16 cos (x) - 16 cos^{2} (x) + 9 - 9 cos^{2} (x)

Simplify

- 4 + 16 cos (x) - 16 cos^{2} (x) + 9 - 9 cos^{2} (x) : 16 cos (x) - 25 cos^{2} (x) + 5

- 4 + 16 cos (x) - 16 cos^{2} (x) + 9 - 9 cos^{2} (x)

Group like terms

= 16 cos (x) - 16 cos^{2} (x) - 9 cos^{2} (x) - 4 + 9

Add similar elements:

- 16 cos^{2} (x) - 9 cos^{2} (x) = - 25 cos^{2} (x)

= 16 cos (x) - 25 cos^{2} (x) - 4 + 9

Add/Subtract the numbers:

- 4 + 9 = 5

= 16 cos (x) - 25 cos^{2} (x) + 5

= 16 cos (x) - 25 cos^{2} (x) + 5

= 16 cos (x) - 25 cos^{2} (x) + 5

5 + 16 cos (x) - 25 cos^{2} (x) = 0

Solve by substitution

5 + 16 cos (x) - 25 cos^{2} (x) = 0

Let:

cos (x) = u

5 + 16 u - 25 u^{2} = 0

5 + 16 u - 25 u^{2} = 0 : u = - \frac{- 8 + 3 21}{25}, u = \frac{8 + 3 21}{25}

5 + 16 u - 25 u^{2} = 0

Write in the standard form

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

- 25 u^{2} + 16 u + 5 = 0

Solve with the quadratic formula

- 25 u^{2} + 16 u + 5 = 0

Quadratic Equation Formula:

For

a = - 25, b = 16, c = 5

u_{1, 2} = \frac{- 16 \pm 1 6 ^{2} - 4 ( - 25 ) \cdot 5}{2 ( - 25 )}

u_{1, 2} = \frac{- 16 \pm 1 6 ^{2} - 4 ( - 25 ) \cdot 5}{2 ( - 25 )}

1 6^{2} - 4 (- 25) \cdot 5 = 621

1 6^{2} - 4 (- 25) \cdot 5

Apply rule

- (- a) = a

= 1 6^{2} + 4 \cdot 25 \cdot 5

Multiply the numbers:

4 \cdot 25 \cdot 5 = 500

= 1 6^{2} + 500

1 6^{2} = 256

= 256 + 500

Add the numbers:

256 + 500 = 756

= 756

Prime factorization of

756 : 2^{2} \cdot 3^{3} \cdot 7

756

756

divides by

2756 = 378 \cdot 2

= 2 \cdot 378

378

divides by

2378 = 189 \cdot 2

= 2 \cdot 2 \cdot 189

189

divides by

3189 = 63 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 63

63

divides by

363 = 21 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 21

21

divides by

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 7

2, 3, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 7

= 2^{2} \cdot 3^{3} \cdot 7

= 3^{3} \cdot 2^{2} \cdot 7

Apply exponent rule:

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

= 2^{2} \cdot 3^{2} \cdot 3 \cdot 7

Apply radical rule:

= 2^{2} 3^{2} 3 \cdot 7

Apply radical rule:

2^{2} = 2

= 2 3^{2} 3 \cdot 7

Apply radical rule:

3^{2} = 3

= 2 \cdot 3 3 \cdot 7

Refine

= 621

u_{1, 2} = \frac{- 16 \pm 6 21}{2 ( - 25 )}

Separate the solutions

u_{1} = \frac{- 16 + 6 21}{2 ( - 25 )}, u_{2} = \frac{- 16 - 6 21}{2 ( - 25 )}

u = \frac{- 16 + 6 21}{2 ( - 25 )} : - \frac{- 8 + 3 21}{25}

\frac{- 16 + 6 21}{2 ( - 25 )}

Remove parentheses:

(- a) = - a

= \frac{- 16 + 6 21}{- 2 \cdot 25}

Multiply the numbers:

2 \cdot 25 = 50

= \frac{- 16 + 6 21}{- 50}

Apply the fraction rule:

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

= - \frac{- 16 + 6 21}{50}

Cancel

\frac{- 16 + 6 21}{50} : \frac{3 21 - 8}{25}

\frac{- 16 + 6 21}{50}

Factor

- 16 + 621 : 2 (- 8 + 321)

- 16 + 621

Rewrite as

= - 2 \cdot 8 + 2 \cdot 321

Factor out common term

2

= 2 (- 8 + 321)

= \frac{2 ( - 8 + 3 21 )}{50}

Cancel the common factor:

2

= \frac{- 8 + 3 21}{25}

= - \frac{3 21 - 8}{25}

= - \frac{- 8 + 3 21}{25}

u = \frac{- 16 - 6 21}{2 ( - 25 )} : \frac{8 + 3 21}{25}

\frac{- 16 - 6 21}{2 ( - 25 )}

Remove parentheses:

(- a) = - a

= \frac{- 16 - 6 21}{- 2 \cdot 25}

Multiply the numbers:

2 \cdot 25 = 50

= \frac{- 16 - 6 21}{- 50}

Apply the fraction rule:

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

- 16 - 621 = - (16 + 621)

= \frac{16 + 6 21}{50}

Factor

16 + 621 : 2 (8 + 321)

16 + 621

Rewrite as

= 2 \cdot 8 + 2 \cdot 321

Factor out common term

2

= 2 (8 + 321)

= \frac{2 ( 8 + 3 21 )}{50}

Cancel the common factor:

2

= \frac{8 + 3 21}{25}

The solutions to the quadratic equation are:

u = - \frac{- 8 + 3 21}{25}, u = \frac{8 + 3 21}{25}

Substitute back

u = cos (x)

cos (x) = - \frac{- 8 + 3 21}{25}, cos (x) = \frac{8 + 3 21}{25}

cos (x) = - \frac{- 8 + 3 21}{25}, cos (x) = \frac{8 + 3 21}{25}

cos (x) = - \frac{- 8 + 3 21}{25} : x = arccos (- \frac{- 8 + 3 21}{25}) + 2 πn, x = - arccos (- \frac{- 8 + 3 21}{25}) + 2 πn

cos (x) = - \frac{- 8 + 3 21}{25}

Apply trig inverse properties

cos (x) = - \frac{- 8 + 3 21}{25}

General solutions for

cos (x) = - \frac{- 8 + 3 21}{25}

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

x = arccos (- \frac{- 8 + 3 21}{25}) + 2 πn, x = - arccos (- \frac{- 8 + 3 21}{25}) + 2 πn

x = arccos (- \frac{- 8 + 3 21}{25}) + 2 πn, x = - arccos (- \frac{- 8 + 3 21}{25}) + 2 πn

cos (x) = \frac{8 + 3 21}{25} : x = arccos (\frac{8 + 3 21}{25}) + 2 πn, x = 2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn

cos (x) = \frac{8 + 3 21}{25}

Apply trig inverse properties

cos (x) = \frac{8 + 3 21}{25}

General solutions for

cos (x) = \frac{8 + 3 21}{25}

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

x = arccos (\frac{8 + 3 21}{25}) + 2 πn, x = 2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn

x = arccos (\frac{8 + 3 21}{25}) + 2 πn, x = 2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 8 + 3 21}{25}) + 2 πn, x = - arccos (- \frac{- 8 + 3 21}{25}) + 2 πn, x = arccos (\frac{8 + 3 21}{25}) + 2 πn, x = 2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

3 sin (x) + 4 cos (x) = 2

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

Check the solution

arccos (- \frac{- 8 + 3 21}{25}) + 2 πn :

True

arccos (- \frac{- 8 + 3 21}{25}) + 2 πn

Plug in

n = 1

arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1

For

3 sin (x) + 4 cos (x) = 2

plug in

x = arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1

3 sin (arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1) + 4 cos (arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

Check the solution

- arccos (- \frac{- 8 + 3 21}{25}) + 2 πn :

False

- arccos (- \frac{- 8 + 3 21}{25}) + 2 πn

Plug in

n = 1

- arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1

For

3 sin (x) + 4 cos (x) = 2

plug in

x = - arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1

3 sin (- arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1) + 4 cos (- arccos (- \frac{- 8 + 3 21}{25}) + 2 π 1) = 2

Refine

- 3.83927 \dots = 2

\Rightarrow False

Check the solution

arccos (\frac{8 + 3 21}{25}) + 2 πn :

False

arccos (\frac{8 + 3 21}{25}) + 2 πn

Plug in

n = 1

arccos (\frac{8 + 3 21}{25}) + 2 π 1

For

3 sin (x) + 4 cos (x) = 2

plug in

x = arccos (\frac{8 + 3 21}{25}) + 2 π 1

3 sin (arccos (\frac{8 + 3 21}{25}) + 2 π 1) + 4 cos (arccos (\frac{8 + 3 21}{25}) + 2 π 1) = 2

Refine

4.95927 \dots = 2

\Rightarrow False

Check the solution

2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn :

True

2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{8 + 3 21}{25}) + 2 π 1

For

3 sin (x) + 4 cos (x) = 2

plug in

x = 2 π - arccos (\frac{8 + 3 21}{25}) + 2 π 1

3 sin (2 π - arccos (\frac{8 + 3 21}{25}) + 2 π 1) + 4 cos (2 π - arccos (\frac{8 + 3 21}{25}) + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

x = arccos (- \frac{- 8 + 3 21}{25}) + 2 πn, x = 2 π - arccos (\frac{8 + 3 21}{25}) + 2 πn

Show solutions in decimal form

x = 1.80278 \dots + 2 πn, x = 2 π - 0.51577 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3*sin(x)+4*cos(x)=2 ?
The general solution for 3*sin(x)+4*cos(x)=2 is x=1.80278…+2pin,x=2pi-0.51577…+2pin