What is the general solution for -3sqrt(3)sin(v)+3cos(v)=3sqrt(2) ?

The general solution for -3sqrt(3)sin(v)+3cos(v)=3sqrt(2) is v=pi+1.30899…+2pin,v=-0.26179…+2pin

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-3sqrt(3)sin(v)+3cos(v)=3sqrt(2)

Solution

- 33 sin (v) + 3 cos (v) = 32

Solution

v = π + 1.30899 \dots + 2 πn, v = - 0.26179 \dots + 2 πn

Degrees

v = 25 5^{\circ} + 36 0^{\circ} n, v = - 1 5^{\circ} + 36 0^{\circ} n

Solution steps

- 33 sin (v) + 3 cos (v) = 32

Add

33 sin (v)

to both sides

3 cos (v) = 32 + 33 sin (v)

Square both sides

(3 cos (v))^{2} = (32 + 33 sin (v))^{2}

Subtract

(32 + 33 sin (v))^{2}

from both sides

9 cos^{2} (v) - 18 - 186 sin (v) - 27 sin^{2} (v) = 0

Rewrite using trig identities

- 18 - 27 sin^{2} (v) + 9 cos^{2} (v) - 18 sin (v) 6

Use the Pythagorean identity:

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

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

= - 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 18 sin (v) 6

Simplify

- 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 18 sin (v) 6 : - 36 sin^{2} (v) - 186 sin (v) - 9

- 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 18 sin (v) 6

= - 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 186 sin (v)

Expand

9 (1 - sin^{2} (v)) : 9 - 9 sin^{2} (v)

9 (1 - sin^{2} (v))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = sin^{2} (v)

= 9 \cdot 1 - 9 sin^{2} (v)

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 sin^{2} (v)

= - 18 - 27 sin^{2} (v) + 9 - 9 sin^{2} (v) - 18 sin (v) 6

Simplify

- 18 - 27 sin^{2} (v) + 9 - 9 sin^{2} (v) - 18 sin (v) 6 : - 36 sin^{2} (v) - 186 sin (v) - 9

- 18 - 27 sin^{2} (v) + 9 - 9 sin^{2} (v) - 18 sin (v) 6

Group like terms

= - 27 sin^{2} (v) - 9 sin^{2} (v) - 186 sin (v) - 18 + 9

Add similar elements:

- 27 sin^{2} (v) - 9 sin^{2} (v) = - 36 sin^{2} (v)

= - 36 sin^{2} (v) - 186 sin (v) - 18 + 9

Add/Subtract the numbers:

- 18 + 9 = - 9

= - 36 sin^{2} (v) - 186 sin (v) - 9

= - 36 sin^{2} (v) - 186 sin (v) - 9

= - 36 sin^{2} (v) - 186 sin (v) - 9

- 9 - 36 sin^{2} (v) - 18 sin (v) 6 = 0

Solve by substitution

- 9 - 36 sin^{2} (v) - 18 sin (v) 6 = 0

Let:

sin (v) = u

- 9 - 36 u^{2} - 18 u 6 = 0

- 9 - 36 u^{2} - 18 u 6 = 0 : u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

- 9 - 36 u^{2} - 18 u 6 = 0

Write in the standard form

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

- 36 u^{2} - 186 u - 9 = 0

Solve with the quadratic formula

- 36 u^{2} - 186 u - 9 = 0

Quadratic Equation Formula:

For

a = - 36, b = - 186, c = - 9

u_{1, 2} = \frac{- ( - 18 6 ) \pm ( - 18 6 ) ^{2} - 4 ( - 36 ) ( - 9 )}{2 ( - 36 )}

u_{1, 2} = \frac{- ( - 18 6 ) \pm ( - 18 6 ) ^{2} - 4 ( - 36 ) ( - 9 )}{2 ( - 36 )}

(- 186)^{2} - 4 (- 36) (- 9) = 182

(- 186)^{2} - 4 (- 36) (- 9)

Apply rule

- (- a) = a

= (- 186)^{2} - 4 \cdot 36 \cdot 9

(- 186)^{2} = 1 8^{2} \cdot 6

(- 186)^{2}

Apply exponent rule:

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

n

is even

(- 186)^{2} = (186)^{2}

= (186)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 1 8^{2} (6)^{2}

(6)^{2} : 6

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 6

= 1 8^{2} \cdot 6

4 \cdot 36 \cdot 9 = 1296

4 \cdot 36 \cdot 9

Multiply the numbers:

4 \cdot 36 \cdot 9 = 1296

= 1296

= 1 8^{2} \cdot 6 - 1296

1 8^{2} \cdot 6 = 1944

1 8^{2} \cdot 6

1 8^{2} = 324

= 324 \cdot 6

Multiply the numbers:

324 \cdot 6 = 1944

= 1944

= 1944 - 1296

Subtract the numbers:

1944 - 1296 = 648

= 648

Prime factorization of

648 : 2^{3} \cdot 3^{4}

648

648

divides by

2648 = 324 \cdot 2

= 2 \cdot 324

324

divides by

2324 = 162 \cdot 2

= 2 \cdot 2 \cdot 162

162

divides by

2162 = 81 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 81

81

divides by

381 = 27 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 27

27

divides by

327 = 9 \cdot 3

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

9

divides by

39 = 3 \cdot 3

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

2, 3

are all prime numbers, therefore no further factorization is possible

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

= 2^{3} \cdot 3^{4}

= 3^{4} \cdot 2^{3}

Apply exponent rule:

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

= 3^{4} \cdot 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2} 3^{4}

Apply radical rule:

2^{2} = 2

= 22 3^{4}

Apply radical rule:

3^{4} = 3^{\frac{4}{2}} = 3^{2}

= 3^{2} \cdot 22

Refine

= 182

u_{1, 2} = \frac{- ( - 18 6 ) \pm 18 2}{2 ( - 36 )}

Separate the solutions

u_{1} = \frac{- ( - 18 6 ) + 18 2}{2 ( - 36 )}, u_{2} = \frac{- ( - 18 6 ) - 18 2}{2 ( - 36 )}

u = \frac{- ( - 18 6 ) + 18 2}{2 ( - 36 )} : - \frac{6 + 2}{4}

\frac{- ( - 18 6 ) + 18 2}{2 ( - 36 )}

Remove parentheses:

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

= \frac{18 6 + 18 2}{- 2 \cdot 36}

Multiply the numbers:

2 \cdot 36 = 72

= \frac{18 6 + 18 2}{- 72}

Apply the fraction rule:

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

= - \frac{18 6 + 18 2}{72}

Cancel

\frac{18 6 + 18 2}{72} : \frac{6 + 2}{4}

\frac{18 6 + 18 2}{72}

Factor out common term

18

= \frac{18 ( 6 + 2 )}{72}

Cancel the common factor:

18

= \frac{6 + 2}{4}

= - \frac{6 + 2}{4}

u = \frac{- ( - 18 6 ) - 18 2}{2 ( - 36 )} : - \frac{6 - 2}{4}

\frac{- ( - 18 6 ) - 18 2}{2 ( - 36 )}

Remove parentheses:

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

= \frac{18 6 - 18 2}{- 2 \cdot 36}

Multiply the numbers:

2 \cdot 36 = 72

= \frac{18 6 - 18 2}{- 72}

Apply the fraction rule:

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

= - \frac{18 6 - 18 2}{72}

Cancel

\frac{18 6 - 18 2}{72} : \frac{6 - 2}{4}

\frac{18 6 - 18 2}{72}

Factor out common term

18

= \frac{18 ( 6 - 2 )}{72}

Cancel the common factor:

18

= \frac{6 - 2}{4}

= - \frac{6 - 2}{4}

The solutions to the quadratic equation are:

u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

Substitute back

u = sin (v)

sin (v) = - \frac{6 + 2}{4}, sin (v) = - \frac{6 - 2}{4}

sin (v) = - \frac{6 + 2}{4}, sin (v) = - \frac{6 - 2}{4}

sin (v) = - \frac{6 + 2}{4} : v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn

sin (v) = - \frac{6 + 2}{4}

Apply trig inverse properties

sin (v) = - \frac{6 + 2}{4}

General solutions for

sin (v) = - \frac{6 + 2}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn

v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn

sin (v) = - \frac{6 - 2}{4} : v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

sin (v) = - \frac{6 - 2}{4}

Apply trig inverse properties

sin (v) = - \frac{6 - 2}{4}

General solutions for

sin (v) = - \frac{6 - 2}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

Combine all the solutions

v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn, v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

- 33 sin (v) + 3 cos (v) = 32

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

Check the solution

arcsin (- \frac{6 + 2}{4}) + 2 πn :

False

arcsin (- \frac{6 + 2}{4}) + 2 πn

Plug in

n = 1

arcsin (- \frac{6 + 2}{4}) + 2 π 1

For

- 33 sin (v) + 3 cos (v) = 32

plug in

v = arcsin (- \frac{6 + 2}{4}) + 2 π 1

- 33 sin (arcsin (- \frac{6 + 2}{4}) + 2 π 1) + 3 cos (arcsin (- \frac{6 + 2}{4}) + 2 π 1) = 32

Refine

5.79555 \dots = 4.24264 \dots

\Rightarrow False

Check the solution

π + arcsin (\frac{6 + 2}{4}) + 2 πn :

True

π + arcsin (\frac{6 + 2}{4}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{6 + 2}{4}) + 2 π 1

For

- 33 sin (v) + 3 cos (v) = 32

plug in

v = π + arcsin (\frac{6 + 2}{4}) + 2 π 1

- 33 sin (π + arcsin (\frac{6 + 2}{4}) + 2 π 1) + 3 cos (π + arcsin (\frac{6 + 2}{4}) + 2 π 1) = 32

Refine

4.24264 \dots = 4.24264 \dots

\Rightarrow True

Check the solution

arcsin (- \frac{6 - 2}{4}) + 2 πn :

True

arcsin (- \frac{6 - 2}{4}) + 2 πn

Plug in

n = 1

arcsin (- \frac{6 - 2}{4}) + 2 π 1

For

- 33 sin (v) + 3 cos (v) = 32

plug in

v = arcsin (- \frac{6 - 2}{4}) + 2 π 1

- 33 sin (arcsin (- \frac{6 - 2}{4}) + 2 π 1) + 3 cos (arcsin (- \frac{6 - 2}{4}) + 2 π 1) = 32

Refine

4.24264 \dots = 4.24264 \dots

\Rightarrow True

Check the solution

π + arcsin (\frac{6 - 2}{4}) + 2 πn :

False

π + arcsin (\frac{6 - 2}{4}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{6 - 2}{4}) + 2 π 1

For

- 33 sin (v) + 3 cos (v) = 32

plug in

v = π + arcsin (\frac{6 - 2}{4}) + 2 π 1

- 33 sin (π + arcsin (\frac{6 - 2}{4}) + 2 π 1) + 3 cos (π + arcsin (\frac{6 - 2}{4}) + 2 π 1) = 32

Refine

- 1.55291 \dots = 4.24264 \dots

\Rightarrow False

v = π + arcsin (\frac{6 + 2}{4}) + 2 πn, v = arcsin (- \frac{6 - 2}{4}) + 2 πn

Show solutions in decimal form

v = π + 1.30899 \dots + 2 πn, v = - 0.26179 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for -3sqrt(3)sin(v)+3cos(v)=3sqrt(2) ?
The general solution for -3sqrt(3)sin(v)+3cos(v)=3sqrt(2) is v=pi+1.30899…+2pin,v=-0.26179…+2pin