Algebra Examples

${(- 3 \cos (x) - 5 \sin (x))}^{2} - 16 \sin^{2} (x) = \frac{18 + 15 \sqrt{3}}{2}$

Step 1

Subtract $\frac{18 + 15 \sqrt{3}}{2}$ from both sides of the equation.

${(- 3 \cos (x) - 5 \sin (x))}^{2} - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2

Simplify ${(- 3 \cos (x) - 5 \sin (x))}^{2} - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2}$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite ${(- 3 \cos (x) - 5 \sin (x))}^{2}$ as $(- 3 \cos (x) - 5 \sin (x)) (- 3 \cos (x) - 5 \sin (x))$ .

$(- 3 \cos (x) - 5 \sin (x)) (- 3 \cos (x) - 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.2

Expand $(- 3 \cos (x) - 5 \sin (x)) (- 3 \cos (x) - 5 \sin (x))$ using the FOIL Method.

Tap for more steps...

Step 2.1.2.1

Apply the distributive property.

$- 3 \cos (x) (- 3 \cos (x) - 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x) - 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.2.2

Apply the distributive property.

$- 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x) - 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.2.3

Apply the distributive property.

$- 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.3.1.1

Multiply $- 3 \cos (x) (- 3 \cos (x))$ .

Tap for more steps...

Step 2.1.3.1.1.1

Multiply $- 3$ by $- 3$ .

$9 \cos (x) \cos (x) - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.1.2

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

$9 (\cos^{1} (x) \cos (x)) - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.1.3

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

$9 (\cos^{1} (x) \cos^{1} (x)) - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.1.4

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

$9 {\cos (x)}^{1 + 1} - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.1.5

Add $1$ and $1$ .

$9 \cos^{2} (x) - 3 \cos (x) (- 5 \sin (x)) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.2

Multiply $- 5$ by $- 3$ .

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) - 5 \sin (x) (- 3 \cos (x)) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.3

Multiply $- 3$ by $- 5$ .

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \sin (x) \cos (x) - 5 \sin (x) (- 5 \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.4

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

Tap for more steps...

Step 2.1.3.1.4.1

Multiply $- 5$ by $- 5$ .

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \sin (x) \cos (x) + 25 \sin (x) \sin (x) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.4.2

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

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \sin (x) \cos (x) + 25 (\sin^{1} (x) \sin (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.4.3

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

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \sin (x) \cos (x) + 25 (\sin^{1} (x) \sin^{1} (x)) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.4.4

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

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \sin (x) \cos (x) + 25 {\sin (x)}^{1 + 1} - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.1.4.5

Add $1$ and $1$ .

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \sin (x) \cos (x) + 25 \sin^{2} (x) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.2

Reorder the factors of $15 \sin (x) \cos (x)$ .

$9 \cos^{2} (x) + 15 \cos (x) \sin (x) + 15 \cos (x) \sin (x) + 25 \sin^{2} (x) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.1.3.3

Add $15 \cos (x) \sin (x)$ and $15 \cos (x) \sin (x)$ .

$9 \cos^{2} (x) + 30 \cos (x) \sin (x) + 25 \sin^{2} (x) - 16 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.2

Subtract $16 \sin^{2} (x)$ from $25 \sin^{2} (x)$ .

$9 \cos^{2} (x) + 30 \cos (x) \sin (x) + 9 \sin^{2} (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.3

Move $9 \sin^{2} (x)$ .

$9 \cos^{2} (x) + 9 \sin^{2} (x) + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.4

Factor $9$ out of $9 \cos^{2} (x)$ .

$9 (\cos^{2} (x)) + 9 \sin^{2} (x) + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.5

Factor $9$ out of $9 \sin^{2} (x)$ .

$9 (\cos^{2} (x)) + 9 (\sin^{2} (x)) + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.6

Factor $9$ out of $9 (\cos^{2} (x)) + 9 (\sin^{2} (x))$ .

$9 (\cos^{2} (x) + \sin^{2} (x)) + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.7

Rearrange terms.

$9 (\sin^{2} (x) + \cos^{2} (x)) + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.8

Apply pythagorean identity.

$9 \cdot 1 + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.9

Multiply $9$ by $1$ .

$9 + 30 \cos (x) \sin (x) - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.10

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$30 \cos (x) \sin (x) + 9 \cdot \frac{2}{2} - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.11

Combine $9$ and $\frac{2}{2}$ .

$30 \cos (x) \sin (x) + \frac{9 \cdot 2}{2} - \frac{18 + 15 \sqrt{3}}{2} = 0$

Step 2.12

Combine the numerators over the common denominator.

$30 \cos (x) \sin (x) + \frac{9 \cdot 2 - (18 + 15 \sqrt{3})}{2} = 0$

Step 2.13

Multiply $9$ by $2$ .

$30 \cos (x) \sin (x) + \frac{18 - (18 + 15 \sqrt{3})}{2} = 0$

Step 2.14

Simplify each term.

Tap for more steps...

Step 2.14.1

Simplify the numerator.

Tap for more steps...

Step 2.14.1.1

Apply the distributive property.

$30 \cos (x) \sin (x) + \frac{18 - 1 \cdot 18 - (15 \sqrt{3})}{2} = 0$

Step 2.14.1.2

Multiply $- 1$ by $18$ .

$30 \cos (x) \sin (x) + \frac{18 - 18 - (15 \sqrt{3})}{2} = 0$

Step 2.14.1.3

Multiply $15$ by $- 1$ .

$30 \cos (x) \sin (x) + \frac{18 - 18 - 15 \sqrt{3}}{2} = 0$

Step 2.14.1.4

Subtract $18$ from $18$ .

$30 \cos (x) \sin (x) + \frac{0 - 15 \sqrt{3}}{2} = 0$

Step 2.14.1.5

Subtract $15 \sqrt{3}$ from $0$ .

$30 \cos (x) \sin (x) + \frac{- 15 \sqrt{3}}{2} = 0$

Step 2.14.2

Move the negative in front of the fraction.

$30 \cos (x) \sin (x) - \frac{15 \sqrt{3}}{2} = 0$

Step 3

Divide each term in the equation by $\cos (x)$ .

$\frac{30 \cos (x) \sin (x)}{\cos (x)} + \frac{- \frac{15 \sqrt{3}}{2}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4

Cancel the common factor of $\cos (x)$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{30 \cos (x) \sin (x)}{\cos (x)} + \frac{- \frac{15 \sqrt{3}}{2}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4.2

Divide $30 \sin (x)$ by $1$ .

$30 \sin (x) + \frac{- \frac{15 \sqrt{3}}{2}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$30 \sin (x) - \frac{15 \sqrt{3}}{2} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$30 \sin (x) - \frac{15 \sqrt{3}}{2} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 7

Combine $\sec (x)$ and $\frac{15 \sqrt{3}}{2}$ .

$30 \sin (x) - \frac{\sec (x) (15 \sqrt{3})}{2} = \frac{0}{\cos (x)}$

Step 8

Move $15$ to the left of $\sec (x)$ .

$30 \sin (x) - \frac{15 \sec (x) \sqrt{3}}{2} = \frac{0}{\cos (x)}$

Step 9

Separate fractions.

$30 \sin (x) - \frac{15 \sec (x) \sqrt{3}}{2} = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$30 \sin (x) - \frac{15 \sec (x) \sqrt{3}}{2} = \frac{0}{1} \cdot \sec (x)$

Step 11

Divide $0$ by $1$ .

$30 \sin (x) - \frac{15 \sec (x) \sqrt{3}}{2} = 0 \sec (x)$

Step 12

Multiply $0$ by $\sec (x)$ .

$30 \sin (x) - \frac{15 \sec (x) \sqrt{3}}{2} = 0$

Step 13

Simplify the left side.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Simplify the numerator.

Tap for more steps...

Step 13.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$30 \sin (x) - \frac{15 \frac{1}{\cos (x)} \sqrt{3}}{2} = 0$

Step 13.1.1.2

Combine exponents.

Tap for more steps...

Step 13.1.1.2.1

Combine $15$ and $\frac{1}{\cos (x)}$ .

$30 \sin (x) - \frac{\frac{15}{\cos (x)} \sqrt{3}}{2} = 0$

Step 13.1.1.2.2

Combine $\frac{15}{\cos (x)}$ and $\sqrt{3}$ .

$30 \sin (x) - \frac{\frac{15 \sqrt{3}}{\cos (x)}}{2} = 0$

Step 13.1.2

Multiply the numerator by the reciprocal of the denominator.

$30 \sin (x) - (\frac{15 \sqrt{3}}{\cos (x)} \cdot \frac{1}{2}) = 0$

Step 13.1.3

Multiply $\frac{15 \sqrt{3}}{\cos (x)}$ by $\frac{1}{2}$ .

$30 \sin (x) - \frac{15 \sqrt{3}}{\cos (x) \cdot 2} = 0$

Step 13.1.4

Move $2$ to the left of $\cos (x)$ .

$30 \sin (x) - \frac{15 \sqrt{3}}{2 \cos (x)} = 0$

Step 14

Divide each term in the equation by $\cos (x)$ .

$\frac{30 \sin (x)}{\cos (x)} + \frac{- \frac{15 \sqrt{3}}{2 \cos (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 15

Separate fractions.

$\frac{30}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{- \frac{15 \sqrt{3}}{2 \cos (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 16

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{30}{1} \cdot \tan (x) + \frac{- \frac{15 \sqrt{3}}{2 \cos (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 17

Divide $30$ by $1$ .

$30 \tan (x) + \frac{- \frac{15 \sqrt{3}}{2 \cos (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 18

Multiply the numerator by the reciprocal of the denominator.

$30 \tan (x) - \frac{15 \sqrt{3}}{2 \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$30 \tan (x) - \frac{15 \sqrt{3}}{2 \cos (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 20

Combine $\sec (x)$ and $\frac{15 \sqrt{3}}{2 \cos (x)}$ .

$30 \tan (x) - \frac{\sec (x) (15 \sqrt{3})}{2 \cos (x)} = \frac{0}{\cos (x)}$

Step 21

Separate fractions.

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Step 22

Rewrite $\sec (x)$ in terms of sines and cosines.

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Step 23

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Step 24

Simplify.

Tap for more steps...

Step 24.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Step 24.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Step 24.3

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

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Step 24.4

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

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Step 24.5

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

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Step 24.6

Add $1$ and $1$ .

$30 \tan (x) - (\frac{15 \sqrt{3}}{2} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Step 25

Combine $\frac{15 \sqrt{3}}{2}$ and $\sec^{2} (x)$ .

$30 \tan (x) - \frac{15 \sqrt{3} \sec^{2} (x)}{2} = \frac{0}{\cos (x)}$

Step 26

Separate fractions.

$30 \tan (x) - \frac{15 \sqrt{3} \sec^{2} (x)}{2} = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 27

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$30 \tan (x) - \frac{15 \sqrt{3} \sec^{2} (x)}{2} = \frac{0}{1} \cdot \sec (x)$

Step 28

Divide $0$ by $1$ .

$30 \tan (x) - \frac{15 \sqrt{3} \sec^{2} (x)}{2} = 0 \sec (x)$

Step 29

Multiply $0$ by $\sec (x)$ .

$30 \tan (x) - \frac{15 \sqrt{3} \sec^{2} (x)}{2} = 0$

Step 30

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$\frac{30 \tan (x) \sqrt{3} (1 + \tan^{2} (x))}{2} = 0$

Step 31

Cancel the common factor of $30$ and $2$ .

Tap for more steps...

Step 31.1

Factor $2$ out of $30 \tan (x) \sqrt{3} (1 + \tan^{2} (x))$ .

$\frac{2 (15 \tan (x) \sqrt{3} (1 + \tan^{2} (x)))}{2} = 0$

Step 31.2

Cancel the common factors.

Tap for more steps...

Step 31.2.1

Factor $2$ out of $2$ .

$\frac{2 (15 \tan (x) \sqrt{3} (1 + \tan^{2} (x)))}{2 (1)} = 0$

Step 31.2.2

Cancel the common factor.

$\frac{2 (15 \tan (x) \sqrt{3} (1 + \tan^{2} (x)))}{2 \cdot 1} = 0$

Step 31.2.3

Rewrite the expression.

$\frac{15 \tan (x) \sqrt{3} (1 + \tan^{2} (x))}{1} = 0$

Step 31.2.4

Divide $15 \tan (x) \sqrt{3} (1 + \tan^{2} (x))$ by $1$ .

$15 \tan (x) \sqrt{3} (1 + \tan^{2} (x)) = 0$

Step 32

Apply the distributive property.

$15 \tan (x) \sqrt{3} \cdot 1 + 15 \tan (x) \sqrt{3} \tan^{2} (x) = 0$

Step 33

Multiply $15$ by $1$ .

$15 \tan (x) \sqrt{3} + 15 \tan (x) \sqrt{3} \tan^{2} (x) = 0$

Step 34

Multiply $\tan (x)$ by $\tan^{2} (x)$ by adding the exponents.

Tap for more steps...

Step 34.1

Move $\tan^{2} (x)$ .

$15 \tan (x) \sqrt{3} + 15 (\tan^{2} (x) \tan (x)) \sqrt{3} = 0$

Step 34.2

Multiply $\tan^{2} (x)$ by $\tan (x)$ .

Tap for more steps...

Step 34.2.1

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

$15 \tan (x) \sqrt{3} + 15 (\tan^{2} (x) \tan (x)) \sqrt{3} = 0$

Step 34.2.2

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

$15 \tan (x) \sqrt{3} + 15 {\tan (x)}^{2 + 1} \sqrt{3} = 0$

Step 34.3

Add $2$ and $1$ .

$15 \tan (x) \sqrt{3} + 15 \tan^{3} (x) \sqrt{3} = 0$

Step 35

Reorder the polynomial.

$15 \tan^{3} (x) \sqrt{3} + 15 \tan (x) \sqrt{3} = 0$

Step 36

Substitute $u$ for $\tan (x)$ .

$15 {(u)}^{3} \sqrt{3} + 15 (u) \sqrt{3} = 0$

Step 37

Factor $15 u \sqrt{3}$ out of $15 u^{3} \sqrt{3} + 15 u \sqrt{3}$ .

Tap for more steps...

Step 37.1

Factor $15 u \sqrt{3}$ out of $15 u^{3} \sqrt{3}$ .

$15 u \sqrt{3} u^{2} + 15 u \sqrt{3} = 0$

Step 37.2

Factor $15 u \sqrt{3}$ out of $15 u \sqrt{3}$ .

$15 u \sqrt{3} u^{2} + 15 u \sqrt{3} \cdot 1 = 0$

Step 37.3

Factor $15 u \sqrt{3}$ out of $15 u \sqrt{3} u^{2} + 15 u \sqrt{3} \cdot 1$ .

$15 u \sqrt{3} (u^{2} + 1) = 0$

Step 38

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

$u = 0$

$u^{2} + 1 = 0$

Step 39

Set $u$ equal to $0$ .

$u = 0$

Step 40

Set $u^{2} + 1$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 40.1

Set $u^{2} + 1$ equal to $0$ .

$u^{2} + 1 = 0$

Step 40.2

Solve $u^{2} + 1 = 0$ for $u$ .

Tap for more steps...

Step 40.2.1

Subtract $1$ from both sides of the equation.

$u^{2} = - 1$

Step 40.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{- 1}$

Step 40.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \pm i$

Step 40.2.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 40.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$u = i$

Step 40.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - i$

Step 40.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = i, - i$

Step 41

The final solution is all the values that make $15 u \sqrt{3} (u^{2} + 1) = 0$ true.

$u = 0, i, - i$

Step 42

Substitute $\tan (x)$ for $u$ .

$\tan (x) = 0, i, - i$

Step 43

Set up each of the solutions to solve for $x$ .

$\tan (x) = 0$

$\tan (x) = i$

$\tan (x) = - i$

Step 44

Solve for $x$ in $\tan (x) = 0$ .

Tap for more steps...

Step 44.1

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

$x = \arctan (0)$

Step 44.2

Simplify the right side.

Tap for more steps...

Step 44.2.1

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

$x = 0$

Step 44.3

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + 0$

Step 44.4

Add $π$ and $0$ .

$x = π$

Step 44.5

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

Tap for more steps...

Step 44.5.1

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

$\frac{π}{| b |}$

Step 44.5.2

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

$\frac{π}{| 1 |}$

Step 44.5.3

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

$\frac{π}{1}$

Step 44.5.4

Divide $π$ by $1$ .

$π$

Step 44.6

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

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

Step 45

Solve for $x$ in $\tan (x) = i$ .

Tap for more steps...

Step 45.1

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

$x = \arctan (i)$

Step 45.2

The inverse tangent of $\arctan (i)$ is undefined.

Undefined

Step 46

Solve for $x$ in $\tan (x) = - i$ .

Tap for more steps...

Step 46.1

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

$x = \arctan (- i)$

Step 46.2

The inverse tangent of $\arctan (- i)$ is undefined.

Undefined

Step 47

List all of the solutions.

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

Step 48

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 49

Exclude the solutions that do not make ${(- 3 \cos (x) - 5 \sin (x))}^{2} - 16 \sin^{2} (x) = \frac{18 + 15 \sqrt{3}}{2}$ true.

No solution