Trigonometry Examples

$4 \cos^{2} (x) - 4 \sqrt{2} \cos (x) + 2 = 0$

Step 1

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

$4 {(u)}^{2} - 4 \sqrt{2} u + 2 = 0$

Step 2

Factor $2$ out of $4 u^{2} - 4 \sqrt{2} u + 2$ .

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Step 2.1

Factor $2$ out of $4 u^{2}$ .

$2 (2 u^{2}) - 4 \sqrt{2} u + 2 = 0$

Step 2.2

Factor $2$ out of $- 4 \sqrt{2} u$ .

$2 (2 u^{2}) + 2 (- 2 \sqrt{2} u) + 2 = 0$

Step 2.3

Factor $2$ out of $2$ .

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

Step 2.4

Factor $2$ out of $2 (2 u^{2}) + 2 (- 2 \sqrt{2} u)$ .

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

Step 2.5

Factor $2$ out of $2 (2 u^{2} - 2 \sqrt{2} u) + 2 (1)$ .

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

Step 3

Divide each term in $2 (2 u^{2} - 2 \sqrt{2} u + 1) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (2 u^{2} - 2 \sqrt{2} u + 1) = 0$ by $2$ .

$\frac{2 (2 u^{2} - 2 \sqrt{2} u + 1)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (2 u^{2} - 2 \sqrt{2} u + 1)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $2 u^{2} - 2 \sqrt{2} u + 1$ by $1$ .

$2 u^{2} - 2 \sqrt{2} u + 1 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$2 u^{2} - 2 \sqrt{2} u + 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 2$ , $b = - 2 \sqrt{2}$ , and $c = 1$ into the quadratic formula and solve for $u$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the product rule to $- 2 \sqrt{2}$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{{(- 2)}^{2} {\sqrt{2}}^{2} - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.2

Raise $- 2$ to the power of $2$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{4 {\sqrt{2}}^{2} - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.3

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

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Step 6.1.3.1

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

$u = \frac{2 \sqrt{2} \pm \sqrt{4 {(2^{\frac{1}{2}})}^{2} - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.3.2

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

$u = \frac{2 \sqrt{2} \pm \sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.3.3

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

$u = \frac{2 \sqrt{2} \pm \sqrt{4 \cdot 2^{\frac{2}{2}} - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.3.4

Cancel the common factor of $2$ .

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Step 6.1.3.4.1

Cancel the common factor.

$u = \frac{2 \sqrt{2} \pm \sqrt{4 \cdot 2^{\frac{2}{2}} - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.3.4.2

Rewrite the expression.

$u = \frac{2 \sqrt{2} \pm \sqrt{4 \cdot 2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.3.5

Evaluate the exponent.

$u = \frac{2 \sqrt{2} \pm \sqrt{4 \cdot 2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.4

Multiply $4$ by $2$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{8 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 6.1.5

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 6.1.5.1

Multiply $- 4$ by $2$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{8 - 8 \cdot 1}}{2 \cdot 2}$

Step 6.1.5.2

Multiply $- 8$ by $1$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{8 - 8}}{2 \cdot 2}$

Step 6.1.6

Subtract $8$ from $8$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{0}}{2 \cdot 2}$

Step 6.1.7

Rewrite $0$ as $0^{2}$ .

$u = \frac{2 \sqrt{2} \pm \sqrt{0^{2}}}{2 \cdot 2}$

Step 6.1.8

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

$u = \frac{2 \sqrt{2} \pm 0}{2 \cdot 2}$

Step 6.1.9

$2 \sqrt{2}$ plus or minus $0$ is $2 \sqrt{2}$ .

$u = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$u = \frac{2 \sqrt{2}}{4}$

Step 6.3

Cancel the common factor of $2$ and $4$ .

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Step 6.3.1

Factor $2$ out of $2 \sqrt{2}$ .

$u = \frac{2 (\sqrt{2})}{4}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $4$ .

$u = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 6.3.2.2

Cancel the common factor.

$u = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 6.3.2.3

Rewrite the expression.

$u = \frac{\sqrt{2}}{2}$

Step 7

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

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 8

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

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 9

Simplify the right side.

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Step 9.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 10

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 11

Simplify $2 π - \frac{π}{4}$ .

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Step 11.1

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 11.2

Combine fractions.

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Step 11.2.1

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 11.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 11.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 12

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

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Step 12.1

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

$\frac{2 π}{| b |}$

Step 12.2

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

$\frac{2 π}{| 1 |}$

Step 12.3

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

$\frac{2 π}{1}$

Step 12.4

Divide $2 π$ by $1$ .

$2 π$

Step 13

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 $n$