Solve for x (6cos(x)-5sin(x))^2+11sin(x)^2=36

${(6 \cos (x) - 5 \sin (x))}^{2} + 11 \sin^{2} (x) = 36$

Subtract $36$ from both sides of the equation.

${(6 \cos (x) - 5 \sin (x))}^{2} + 11 \sin^{2} (x) - 36 = 0$

Simplify ${(6 \cos (x) - 5 \sin (x))}^{2} + 11 \sin^{2} (x) - 36$ .

Simplify each term.

Rewrite ${(6 \cos (x) - 5 \sin (x))}^{2}$ as $(6 \cos (x) - 5 \sin (x)) (6 \cos (x) - 5 \sin (x))$ .

$(6 \cos (x) - 5 \sin (x)) (6 \cos (x) - 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Expand $(6 \cos (x) - 5 \sin (x)) (6 \cos (x) - 5 \sin (x))$ using the FOIL Method.

Apply the distributive property.

$6 \cos (x) (6 \cos (x) - 5 \sin (x)) - 5 \sin (x) (6 \cos (x) - 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Apply the distributive property.

$6 \cos (x) (6 \cos (x)) + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x) - 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Apply the distributive property.

$6 \cos (x) (6 \cos (x)) + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $6 \cos (x) (6 \cos (x))$ .

Multiply $6$ by $6$ .

$36 \cos (x) \cos (x) + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Raise $\cos (x)$ to the power of $1$ .

$36 (\cos^{1} (x) \cos (x)) + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Raise $\cos (x)$ to the power of $1$ .

$36 (\cos^{1} (x) \cos^{1} (x)) + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$36 {\cos (x)}^{1 + 1} + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Add $1$ and $1$ .

$36 \cos^{2} (x) + 6 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Multiply $- 5$ by $6$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 5 \sin (x) (6 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Multiply $6$ by $- 5$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \sin (x) \cos (x) - 5 \sin (x) (- 5 \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Multiply $- 5 \sin (x) (- 5 \sin (x))$ .

Multiply $- 5$ by $- 5$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \sin (x) \cos (x) + 25 \sin (x) \sin (x) + 11 \sin^{2} (x) - 36 = 0$

Raise $\sin (x)$ to the power of $1$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \sin (x) \cos (x) + 25 (\sin^{1} (x) \sin (x)) + 11 \sin^{2} (x) - 36 = 0$

Raise $\sin (x)$ to the power of $1$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \sin (x) \cos (x) + 25 (\sin^{1} (x) \sin^{1} (x)) + 11 \sin^{2} (x) - 36 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \sin (x) \cos (x) + 25 {\sin (x)}^{1 + 1} + 11 \sin^{2} (x) - 36 = 0$

Add $1$ and $1$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \sin (x) \cos (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) - 36 = 0$

Reorder the factors of $- 30 \sin (x) \cos (x)$ .

$36 \cos^{2} (x) - 30 \cos (x) \sin (x) - 30 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) - 36 = 0$

Subtract $30 \cos (x) \sin (x)$ from $- 30 \cos (x) \sin (x)$ .

$36 \cos^{2} (x) - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) - 36 = 0$

Move $- 36$ .

$36 \cos^{2} (x) - 36 - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Reorder $36 \cos^{2} (x)$ and $- 36$ .

$- 36 + 36 \cos^{2} (x) - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Factor $- 36$ out of $- 36$ .

$- 36 (1) + 36 \cos^{2} (x) - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Factor $- 36$ out of $36 \cos^{2} (x)$ .

$- 36 (1) - 36 (- \cos^{2} (x)) - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Factor $- 36$ out of $- 36 (1) - 36 (- \cos^{2} (x))$ .

$- 36 (1 - \cos^{2} (x)) - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Apply pythagorean identity.

$- 36 \sin^{2} (x) - 60 \cos (x) \sin (x) + 25 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Add $- 36 \sin^{2} (x)$ and $25 \sin^{2} (x)$ .

$- 60 \cos (x) \sin (x) - 11 \sin^{2} (x) + 11 \sin^{2} (x) = 0$

Combine the opposite terms in $- 60 \cos (x) \sin (x) - 11 \sin^{2} (x) + 11 \sin^{2} (x)$ .

Add $- 11 \sin^{2} (x)$ and $11 \sin^{2} (x)$ .

$- 60 \cos (x) \sin (x) + 0 = 0$

Add $- 60 \cos (x) \sin (x)$ and $0$ .

$- 60 \cos (x) \sin (x) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

$\sin (x) = 0$

Set $\cos (x)$ equal to $0$ and solve for $x$ .

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Solve $\cos (x) = 0$ for $x$ .

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (0)$

Simplify the right side.

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Simplify $2 π - \frac{π}{2}$ .

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Combine fractions.

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Find the period of $\cos (x)$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Set $\sin (x)$ equal to $0$ and solve for $x$ .

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Solve $\sin (x) = 0$ for $x$ .

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (0)$

Simplify the right side.

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Subtract $0$ from $π$ .

$x = π$

Find the period of $\sin (x)$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

The final solution is all the values that make $- 60 \cos (x) \sin (x) = 0$ true.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, 2 π n, π + 2 π n$ , for any integer $n$

Consolidate the answers.

$x = \frac{π n}{2}$ , for any integer $n$

Solve for x (6cos(x)-5sin(x))^2+11sin(x)^2=36

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