Trigonometry Examples

$2 \cos^{2} (x) - 2 \cos (x) - 1 = 0$

Step 1

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

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

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

Step 4.1.2.2

Multiply $- 8$ by $- 1$ .

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

Step 4.1.3

Add $4$ and $8$ .

$u = \frac{2 \pm \sqrt{12}}{2 \cdot 2}$

Step 4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.4.1

Factor $4$ out of $12$ .

$u = \frac{2 \pm \sqrt{4 (3)}}{2 \cdot 2}$

Step 4.1.4.2

Rewrite $4$ as $2^{2}$ .

$u = \frac{2 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical.

$u = \frac{2 \pm 2 \sqrt{3}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$u = \frac{2 \pm 2 \sqrt{3}}{4}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{3}}{4}$ .

$u = \frac{1 \pm \sqrt{3}}{2}$

Step 5

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}$

Step 6

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

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

Step 7

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

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

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

Step 8

Solve for $x$ in $\cos (x) = \frac{1 + \sqrt{3}}{2}$ .

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Step 8.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $\frac{1 + \sqrt{3}}{2}$ does not fall in this range, there is no solution.

No solution

Step 9

Solve for $x$ in $\cos (x) = \frac{1 - \sqrt{3}}{2}$ .

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Step 9.1

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

$x = \arccos (\frac{1 - \sqrt{3}}{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

Evaluate $\arccos (\frac{1 - \sqrt{3}}{2})$ .

$x = 1.94553075$

Step 9.3

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.

$x = 2 (3.14159265) - 1.94553075$

Step 9.4

Solve for $x$ .

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Step 9.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.94553075$

Step 9.4.2

Simplify $2 (3.14159265) - 1.94553075$ .

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Step 9.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.94553075$

Step 9.4.2.2

Subtract $1.94553075$ from $6.2831853$ .

$x = 4.33765454$

Step 9.5

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

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Step 9.5.1

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

$\frac{2 π}{| b |}$

Step 9.5.2

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

$\frac{2 π}{| 1 |}$

Step 9.5.3

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

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

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

$x = 1.94553075 + 2 π n, 4.33765454 + 2 π n$ , for any integer $n$

Step 10

List all of the solutions.

$x = 1.94553075 + 2 π n, 4.33765454 + 2 π n$ , for any integer $n$