Trigonometry Examples

$\sec^{2} (x) + \sqrt{3} \sec (x) - \sqrt{2} \sec (x) - \sqrt{6} = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

${(\frac{1}{\cos (x)})}^{2} + \sqrt{3} \sec (x) - \sqrt{2} \sec (x) - \sqrt{6} = 0$

Step 1.1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{1^{2}}{\cos^{2} (x)} + \sqrt{3} \sec (x) - \sqrt{2} \sec (x) - \sqrt{6} = 0$

Step 1.1.3

One to any power is one.

$\frac{1}{\cos^{2} (x)} + \sqrt{3} \sec (x) - \sqrt{2} \sec (x) - \sqrt{6} = 0$

Step 1.1.4

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

$\frac{1}{\cos^{2} (x)} + \sqrt{3} \frac{1}{\cos (x)} - \sqrt{2} \sec (x) - \sqrt{6} = 0$

Step 1.1.5

Combine $\sqrt{3}$ and $\frac{1}{\cos (x)}$ .

$\frac{1}{\cos^{2} (x)} + \frac{\sqrt{3}}{\cos (x)} - \sqrt{2} \sec (x) - \sqrt{6} = 0$

Step 1.1.6

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

$\frac{1}{\cos^{2} (x)} + \frac{\sqrt{3}}{\cos (x)} - \sqrt{2} \frac{1}{\cos (x)} - \sqrt{6} = 0$

Step 1.1.7

Combine $\frac{1}{\cos (x)}$ and $\sqrt{2}$ .

$\frac{1}{\cos^{2} (x)} + \frac{\sqrt{3}}{\cos (x)} - \frac{\sqrt{2}}{\cos (x)} - \sqrt{6} = 0$

Step 2

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{1}{\cos^{2} (x)} + \frac{\sqrt{3}}{\cos (x)} - \frac{\sqrt{2}}{\cos (x)} - \sqrt{6}) = \cos (x) \cdot 0$

Step 3

Apply the distributive property.

$\cos (x) \frac{1}{\cos^{2} (x)} + \cos (x) \frac{\sqrt{3}}{\cos (x)} + \cos (x) (- \frac{\sqrt{2}}{\cos (x)}) + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$\cos (x) \frac{1}{\cos (x) \cos (x)} + \cos (x) \frac{\sqrt{3}}{\cos (x)} + \cos (x) (- \frac{\sqrt{2}}{\cos (x)}) + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 4.1.2

Cancel the common factor.

$\cos (x) \frac{1}{\cos (x) \cos (x)} + \cos (x) \frac{\sqrt{3}}{\cos (x)} + \cos (x) (- \frac{\sqrt{2}}{\cos (x)}) + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 4.1.3

Rewrite the expression.

$\frac{1}{\cos (x)} + \cos (x) \frac{\sqrt{3}}{\cos (x)} + \cos (x) (- \frac{\sqrt{2}}{\cos (x)}) + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

$\frac{1}{\cos (x)} + \cos (x) \frac{\sqrt{3}}{\cos (x)} + \cos (x) (- \frac{\sqrt{2}}{\cos (x)}) + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 4.2.2

Rewrite the expression.

$\frac{1}{\cos (x)} + \sqrt{3} + \cos (x) (- \frac{\sqrt{2}}{\cos (x)}) + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 4.3

Rewrite using the commutative property of multiplication.

$\frac{1}{\cos (x)} + \sqrt{3} - \cos (x) \frac{\sqrt{2}}{\cos (x)} + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 5

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

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Step 5.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\frac{1}{\cos (x)} + \sqrt{3} + \cos (x) \cdot - 1 \frac{\sqrt{2}}{\cos (x)} + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 5.2

Cancel the common factor.

$\frac{1}{\cos (x)} + \sqrt{3} + \cos (x) \cdot - 1 \frac{\sqrt{2}}{\cos (x)} + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 5.3

Rewrite the expression.

$\frac{1}{\cos (x)} + \sqrt{3} - \sqrt{2} + \cos (x) (- \sqrt{6}) = \cos (x) \cdot 0$

Step 6

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

$\frac{1}{\cos (x)} + \sqrt{3} - \sqrt{2} + \cos (x) (- \sqrt{6}) = 0$

Step 7

Reorder factors in $\frac{1}{\cos (x)} + \sqrt{3} - \sqrt{2} + \cos (x) (- \sqrt{6}) = 0$ .

$\frac{1}{\cos (x)} + \sqrt{3} - \sqrt{2} - \cos (x) \sqrt{6} = 0$

Step 8

Find the LCD of the terms in the equation.

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Step 8.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\cos (x), 1, 1, 1, 1$

Step 8.2

The LCM of one and any expression is the expression.

$\cos (x)$

Step 9

Multiply each term in $\frac{1}{\cos (x)} + \sqrt{3} - \sqrt{2} - \cos (x) \sqrt{6} = 0$ by $\cos (x)$ to eliminate the fractions.

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Step 9.1

Multiply each term in $\frac{1}{\cos (x)} + \sqrt{3} - \sqrt{2} - \cos (x) \sqrt{6} = 0$ by $\cos (x)$ .

$\frac{1}{\cos (x)} \cos (x) + \sqrt{3} \cos (x) - \sqrt{2} \cos (x) - \cos (x) \sqrt{6} \cos (x) = 0 \cos (x)$

Step 9.2

Simplify the left side.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

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

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Step 9.2.1.1.1

Cancel the common factor.

$\frac{1}{\cos (x)} \cos (x) + \sqrt{3} \cos (x) - \sqrt{2} \cos (x) - \cos (x) \sqrt{6} \cos (x) = 0 \cos (x)$

Step 9.2.1.1.2

Rewrite the expression.

$1 + \sqrt{3} \cos (x) - \sqrt{2} \cos (x) - \cos (x) \sqrt{6} \cos (x) = 0 \cos (x)$

Step 9.2.1.2

Multiply $\cos (x)$ by $\cos (x)$ by adding the exponents.

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Step 9.2.1.2.1

Move $\cos (x)$ .

$1 + \sqrt{3} \cos (x) - \sqrt{2} \cos (x) - (\cos (x) \cos (x)) \sqrt{6} = 0 \cos (x)$

Step 9.2.1.2.2

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

$1 + \sqrt{3} \cos (x) - \sqrt{2} \cos (x) - \cos^{2} (x) \sqrt{6} = 0 \cos (x)$

Step 9.3

Simplify the right side.

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Step 9.3.1

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

$1 + \sqrt{3} \cos (x) - \sqrt{2} \cos (x) - \cos^{2} (x) \sqrt{6} = 0$

Step 10

Solve the equation.

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Step 10.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 10.2

Substitute the values $a = - \sqrt{6}$ , $b = \sqrt{3} - \sqrt{2}$ , and $c = 1$ into the quadratic formula and solve for $\cos (x)$ .

$\frac{- (\sqrt{3} - \sqrt{2}) \pm \sqrt{{(\sqrt{3} - \sqrt{2})}^{2} - 4 \cdot (- \sqrt{6} \cdot 1)}}{2 (- \sqrt{6})}$

Step 10.3

Simplify.

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Step 10.3.1

Simplify the numerator.

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Step 10.3.1.1

Apply the distributive property.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{{(\sqrt{3} - \sqrt{2})}^{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.2

Multiply $- - \sqrt{2}$ .

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Step 10.3.1.2.1

Multiply $- 1$ by $- 1$ .

$\cos (x) = \frac{- \sqrt{3} + 1 \sqrt{2} \pm \sqrt{{(\sqrt{3} - \sqrt{2})}^{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.2.2

Multiply $\sqrt{2}$ by $1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{{(\sqrt{3} - \sqrt{2})}^{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.3

Rewrite ${(\sqrt{3} - \sqrt{2})}^{2}$ as $(\sqrt{3} - \sqrt{2}) (\sqrt{3} - \sqrt{2})$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{(\sqrt{3} - \sqrt{2}) (\sqrt{3} - \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.4

Expand $(\sqrt{3} - \sqrt{2}) (\sqrt{3} - \sqrt{2})$ using the FOIL Method.

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Step 10.3.1.4.1

Apply the distributive property.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{\sqrt{3} (\sqrt{3} - \sqrt{2}) - \sqrt{2} (\sqrt{3} - \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.4.2

Apply the distributive property.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{2}) - \sqrt{2} (\sqrt{3} - \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.4.3

Apply the distributive property.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5

Simplify and combine like terms.

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Step 10.3.1.5.1

Simplify each term.

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Step 10.3.1.5.1.1

Combine using the product rule for radicals.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{\sqrt{3 \cdot 3} + \sqrt{3} (- \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.2

Multiply $3$ by $3$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{\sqrt{9} + \sqrt{3} (- \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.3

Rewrite $9$ as $3^{2}$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{\sqrt{3^{2}} + \sqrt{3} (- \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 + \sqrt{3} (- \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.5

Multiply $\sqrt{3} (- \sqrt{2})$ .

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Step 10.3.1.5.1.5.1

Combine using the product rule for radicals.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{3 \cdot 2} - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.5.2

Multiply $3$ by $2$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{2} \sqrt{3} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.6

Multiply $- \sqrt{2} \sqrt{3}$ .

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Step 10.3.1.5.1.6.1

Combine using the product rule for radicals.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{3 \cdot 2} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.6.2

Multiply $3$ by $2$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} - \sqrt{2} (- \sqrt{2}) - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.7

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 10.3.1.5.1.7.1

Multiply $- 1$ by $- 1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + 1 \sqrt{2} \sqrt{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.7.2

Multiply $\sqrt{2}$ by $1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + \sqrt{2} \sqrt{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + \sqrt{2} \sqrt{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.7.4

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + \sqrt{2} \sqrt{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.7.5

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

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + {\sqrt{2}}^{1 + 1} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.7.6

Add $1$ and $1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + {\sqrt{2}}^{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.8

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 10.3.1.5.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + {(2^{\frac{1}{2}})}^{2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + 2^{\frac{1}{2} \cdot 2} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.8.3

Combine $\frac{1}{2}$ and $2$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + 2^{\frac{2}{2}} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.8.4

Cancel the common factor of $2$ .

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Step 10.3.1.5.1.8.4.1

Cancel the common factor.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + 2^{\frac{2}{2}} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.8.4.2

Rewrite the expression.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + 2 - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.1.8.5

Evaluate the exponent.

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{3 - \sqrt{6} - \sqrt{6} + 2 - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.2

Add $3$ and $2$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 - \sqrt{6} - \sqrt{6} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.5.3

Subtract $\sqrt{6}$ from $- \sqrt{6}$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 - 2 \sqrt{6} - 4 \cdot (- 1 \sqrt{6}) \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.6

Multiply $- 4$ by $- 1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 - 2 \sqrt{6} + 4 \sqrt{6} \cdot 1}}{2 (- \sqrt{6})}$

Step 10.3.1.7

Multiply $4$ by $1$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 - 2 \sqrt{6} + 4 \sqrt{6}}}{2 (- \sqrt{6})}$

Step 10.3.1.8

Add $- 2 \sqrt{6}$ and $4 \sqrt{6}$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{2 (- \sqrt{6})}$

Step 10.3.2

Multiply $- 1$ by $2$ .

$\cos (x) = \frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{- 2 \sqrt{6}}$

Step 10.3.3

Simplify $\frac{- \sqrt{3} + \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{- 2 \sqrt{6}}$ .

$\cos (x) = \frac{\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{2 \sqrt{6}}$

Step 10.3.4

Multiply $\frac{\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\cos (x) = \frac{\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 10.3.5

Combine and simplify the denominator.

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Step 10.3.5.1

Multiply $\frac{\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 10.3.5.2

Move $\sqrt{6}$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 10.3.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 10.3.5.4

Raise $\sqrt{6}$ to the power of $1$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 10.3.5.5

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

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 10.3.5.6

Add $1$ and $1$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 10.3.5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 10.3.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 10.3.5.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 10.3.5.7.3

Combine $\frac{1}{2}$ and $2$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 10.3.5.7.4

Cancel the common factor of $2$ .

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Step 10.3.5.7.4.1

Cancel the common factor.

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 10.3.5.7.4.2

Rewrite the expression.

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 \cdot 6}$

Step 10.3.5.7.5

Evaluate the exponent.

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{2 \cdot 6}$

Step 10.3.6

Multiply $2$ by $6$ .

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} \pm \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}$

Step 10.4

The final answer is the combination of both solutions.

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} + \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}, \frac{(\sqrt{3} - \sqrt{2} - \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}$

Step 11

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

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} + \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}$

$\cos (x) = \frac{(\sqrt{3} - \sqrt{2} - \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}$

Step 12

Solve for $x$ in $\cos (x) = \frac{(\sqrt{3} - \sqrt{2} + \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}$ .

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Step 12.1

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

$x = \arccos (\frac{(\sqrt{3} - \sqrt{2} + \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arccos (\frac{(\sqrt{3} - \sqrt{2} + \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12})$ .

$x = \frac{π}{4}$

Step 12.3

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.

$x = 2 π - \frac{π}{4}$

Step 12.4

Simplify $2 π - \frac{π}{4}$ .

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Step 12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 12.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 12.5

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

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Step 12.5.1

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

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 13

Solve for $x$ in $\cos (x) = \frac{(\sqrt{3} - \sqrt{2} - \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12}$ .

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Step 13.1

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

$x = \arccos (\frac{(\sqrt{3} - \sqrt{2} - \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12})$

Step 13.2

Simplify the right side.

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Step 13.2.1

Evaluate $\arccos (\frac{(\sqrt{3} - \sqrt{2} - \sqrt{5 + 2 \sqrt{6}}) \sqrt{6}}{12})$ .

$x = 2.18627603$

Step 13.3

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.

$x = 2 (3.14159265) - 2.18627603$

Step 13.4

Solve for $x$ .

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Step 13.4.1

Remove parentheses.

$x = 2 (3.14159265) - 2.18627603$

Step 13.4.2

Simplify $2 (3.14159265) - 2.18627603$ .

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Step 13.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.18627603$

Step 13.4.2.2

Subtract $2.18627603$ from $6.2831853$ .

$x = 4.09690927$

Step 13.5

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

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Step 13.5.1

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

$\frac{2 π}{| b |}$

Step 13.5.2

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

$\frac{2 π}{| 1 |}$

Step 13.5.3

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

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

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

$x = 2.18627603 + 2 π n, 4.09690927 + 2 π n$ , for any integer $n$

Step 14

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, 2.18627603 + 2 π n, 4.09690927 + 2 π n$ , for any integer $n$