What is the general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 ?

The general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 is x= pi/(12)+pin,x=(5pi)/(12)+pin

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(-6cos(x)-5sin(x))^2+11sin^2(x)=51

Solution

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

Solution

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Degrees

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

Solution steps

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

Subtract

51

from both sides

36 cos^{2} (x) + 60 cos (x) sin (x) + 36 sin^{2} (x) - 51 = 0

Rewrite using trig identities

- 51 + 36 cos^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Use the Pythagorean identity:

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

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

= - 51 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Simplify

- 51 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x) - 15

- 51 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Expand

36 (1 - sin^{2} (x)) : 36 - 36 sin^{2} (x)

36 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 36, b = 1, c = sin^{2} (x)

= 36 \cdot 1 - 36 sin^{2} (x)

Multiply the numbers:

36 \cdot 1 = 36

= 36 - 36 sin^{2} (x)

= - 51 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Simplify

- 51 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x) - 15

- 51 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Add similar elements:

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

= - 51 + 36 + 60 cos (x) sin (x)

Add/Subtract the numbers:

- 51 + 36 = - 15

= 60 cos (x) sin (x) - 15

= 60 cos (x) sin (x) - 15

= 60 cos (x) sin (x) - 15

Use the Double Angle identity:

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

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

= - 15 + 60 \cdot \frac{sin ( 2 x )}{2}

- 15 + 60 \cdot \frac{sin ( 2 x )}{2} = 0

60 \cdot \frac{sin ( 2 x )}{2} = 30 sin (2 x)

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

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 60}{2}

Divide the numbers:

\frac{60}{2} = 30

= 30 sin (2 x)

- 15 + 30 sin (2 x) = 0

Move

15

to the right side

- 15 + 30 sin (2 x) = 0

Add

15

to both sides

- 15 + 30 sin (2 x) + 15 = 0 + 15

Simplify

30 sin (2 x) = 15

30 sin (2 x) = 15

Divide both sides by

30

30 sin (2 x) = 15

Divide both sides by

30

\frac{30 sin ( 2 x )}{30} = \frac{15}{30}

Simplify

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

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

General solutions for

sin (2 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}

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

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

Solve

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{π}{6}}{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}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

Solve

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{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{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 ?
The general solution for (-6cos(x)-5sin(x))^2+11sin^2(x)=51 is x= pi/(12)+pin,x=(5pi)/(12)+pin