What is the general solution for (3cos(x)-6sin(x))^2-27sin^2(x)=-9 ?

The general solution for (3cos(x)-6sin(x))^2-27sin^2(x)=-9 is x= pi/4+pin

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(3cos(x)-6sin(x))^2-27sin^2(x)=-9

Solution

(3 cos (x) - 6 sin (x))^{2} - 27 sin^{2} (x) = - 9

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

(3 cos (x) - 6 sin (x))^{2} - 27 sin^{2} (x) = - 9

Subtract

- 9

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

9 + 9 (1 - sin^{2} (x)) + 9 sin^{2} (x) - 36 cos (x) sin (x) : - 36 cos (x) sin (x) + 18

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

9 \cdot 1 = 9

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

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

Simplify

9 + 9 - 9 sin^{2} (x) + 9 sin^{2} (x) - 36 cos (x) sin (x) : - 36 cos (x) sin (x) + 18

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

Add similar elements:

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

= 9 + 9 - 36 cos (x) sin (x)

Add the numbers:

9 + 9 = 18

= - 36 cos (x) sin (x) + 18

= - 36 cos (x) sin (x) + 18

= - 36 cos (x) sin (x) + 18

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

= 18 - 36 \cdot \frac{sin ( 2 x )}{2}

18 - 36 \cdot \frac{sin ( 2 x )}{2} = 0

36 \cdot \frac{sin ( 2 x )}{2} = 18 sin (2 x)

36 \cdot \frac{sin ( 2 x )}{2}

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 36}{2}

Divide the numbers:

\frac{36}{2} = 18

= 18 sin (2 x)

18 - 18 sin (2 x) = 0

Move

18

to the right side

18 - 18 sin (2 x) = 0

Subtract

18

from both sides

18 - 18 sin (2 x) - 18 = 0 - 18

Simplify

- 18 sin (2 x) = - 18

- 18 sin (2 x) = - 18

Divide both sides by

- 18

- 18 sin (2 x) = - 18

Divide both sides by

- 18

\frac{- 18 sin ( 2 x )}{- 18} = \frac{- 18}{- 18}

Simplify

sin (2 x) = 1

sin (2 x) = 1

General solutions for

sin (2 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}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

Solve

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (3cos(x)-6sin(x))^2-27sin^2(x)=-9 ?
The general solution for (3cos(x)-6sin(x))^2-27sin^2(x)=-9 is x= pi/4+pin