What is the general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 ?

The general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 is x= pi/4+pin

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(6cos(x)+5sin(x))^2+11sin^2(x)=66

Solution

(6 cos (x) + 5 sin (x))^{2} + 11 sin^{2} (x) = 66

Solution

x = \frac{π}{4} + πn

Degrees

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

Solution steps

(6 cos (x) + 5 sin (x))^{2} + 11 sin^{2} (x) = 66

Subtract

66

from both sides

36 cos^{2} (x) + 60 cos (x) sin (x) + 36 sin^{2} (x) - 66 = 0

Rewrite using trig identities

- 66 + 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)

= - 66 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Simplify

- 66 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x) - 30

- 66 + 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)

= - 66 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Simplify

- 66 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x) - 30

- 66 + 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

= - 66 + 36 + 60 cos (x) sin (x)

Add/Subtract the numbers:

- 66 + 36 = - 30

= 60 cos (x) sin (x) - 30

= 60 cos (x) sin (x) - 30

= 60 cos (x) sin (x) - 30

Use the Double Angle identity:

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

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

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

- 30 + 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)

- 30 + 30 sin (2 x) = 0

Move

30

to the right side

- 30 + 30 sin (2 x) = 0

Add

30

to both sides

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

Simplify

30 sin (2 x) = 30

30 sin (2 x) = 30

Divide both sides by

30

30 sin (2 x) = 30

Divide both sides by

30

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

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

Popular Examples

-3sin(B)-2=sin(B)-2

- 3 sin (B) - 2 = sin (B) - 2

sin(θ/2)=sqrt(2)-sin(θ/2)

sin (\frac{θ}{2}) = 2 - sin (\frac{θ}{2})

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

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

(tan(x)+1)(2sin(x)-1)=0

(tan (x) + 1) (2 sin (x) - 1) = 0

sin(θ)=(0.3924)/(cos(θ))

sin (θ) = \frac{0.3924}{cos ( θ )}

Frequently Asked Questions (FAQ)

What is the general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 ?
The general solution for (6cos(x)+5sin(x))^2+11sin^2(x)=66 is x= pi/4+pin