What is the general solution for 6sin(x)-3sin^2(x)=3-cos^2(x) ?

The general solution for 6sin(x)-3sin^2(x)=3-cos^2(x) is x= pi/6+2pin,x=(5pi)/6+2pin,x= pi/2+2pin

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6sin(x)-3sin^2(x)=3-cos^2(x)

Solution

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

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3 - cos^{2} (x)

from both sides

6 sin (x) - 3 sin^{2} (x) - 3 + cos^{2} (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= - 3 + 1 - sin^{2} (x) - 3 sin^{2} (x) + 6 sin (x)

Simplify

- 3 + 1 - sin^{2} (x) - 3 sin^{2} (x) + 6 sin (x) : 6 sin (x) - 4 sin^{2} (x) - 2

- 3 + 1 - sin^{2} (x) - 3 sin^{2} (x) + 6 sin (x)

Add similar elements:

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

= - 3 + 1 - 4 sin^{2} (x) + 6 sin (x)

Add/Subtract the numbers:

- 3 + 1 = - 2

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

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

- 2 - 4 sin^{2} (x) + 6 sin (x) = 0

Solve by substitution

- 2 - 4 sin^{2} (x) + 6 sin (x) = 0

Let:

sin (x) = u

- 2 - 4 u^{2} + 6 u = 0

- 2 - 4 u^{2} + 6 u = 0 : u = \frac{1}{2}, u = 1

- 2 - 4 u^{2} + 6 u = 0

Write in the standard form

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

- 4 u^{2} + 6 u - 2 = 0

Solve with the quadratic formula

- 4 u^{2} + 6 u - 2 = 0

Quadratic Equation Formula:

For

a = - 4, b = 6, c = - 2

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

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

6^{2} - 4 (- 4) (- 2) = 2

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

Apply rule

- (- a) = a

= 6^{2} - 4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

Subtract the numbers:

36 - 32 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- 6 \pm 2}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 6 + 2}{2 ( - 4 )}, u_{2} = \frac{- 6 - 2}{2 ( - 4 )}

u = \frac{- 6 + 2}{2 ( - 4 )} : \frac{1}{2}

\frac{- 6 + 2}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

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

Add/Subtract the numbers:

- 6 + 2 = - 4

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{- 8}

Apply the fraction rule:

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

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{- 6 - 2}{2 ( - 4 )} : 1

\frac{- 6 - 2}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

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

Subtract the numbers:

- 6 - 2 = - 8

= \frac{- 8}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 8}{- 8}

Apply the fraction rule:

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

= \frac{8}{8}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = 1

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = 1

sin (x) = \frac{1}{2}, sin (x) = 1

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin(x)-3sin^2(x)=3-cos^2(x) ?
The general solution for 6sin(x)-3sin^2(x)=3-cos^2(x) is x= pi/6+2pin,x=(5pi)/6+2pin,x= pi/2+2pin