Trigonometry Examples

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

Step 1

Replace the $2 \sin^{2} (x)$ with $2 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $2$ by $1$ .

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

Step 2.3

Multiply $- 1$ by $2$ .

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

Step 3

Add $2$ and $1$ .

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

Step 4

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

$- 2 {(u)}^{2} - 6 u + 3 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 2$ , $b = - 6$ , and $c = 3$ into the quadratic formula and solve for $u$ .

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 6$ to the power of $2$ .

$u = \frac{6 \pm \sqrt{36 - 4 \cdot - 2 \cdot 3}}{2 \cdot - 2}$

Step 7.1.2

Multiply $- 4 \cdot - 2 \cdot 3$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 2$ .

$u = \frac{6 \pm \sqrt{36 + 8 \cdot 3}}{2 \cdot - 2}$

Step 7.1.2.2

Multiply $8$ by $3$ .

$u = \frac{6 \pm \sqrt{36 + 24}}{2 \cdot - 2}$

Step 7.1.3

Add $36$ and $24$ .

$u = \frac{6 \pm \sqrt{60}}{2 \cdot - 2}$

Step 7.1.4

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 7.1.4.1

Factor $4$ out of $60$ .

$u = \frac{6 \pm \sqrt{4 (15)}}{2 \cdot - 2}$

Step 7.1.4.2

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

$u = \frac{6 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot - 2}$

Step 7.1.5

Pull terms out from under the radical.

$u = \frac{6 \pm 2 \sqrt{15}}{2 \cdot - 2}$

Step 7.2

Multiply $2$ by $- 2$ .

$u = \frac{6 \pm 2 \sqrt{15}}{- 4}$

Step 7.3

Simplify $\frac{6 \pm 2 \sqrt{15}}{- 4}$ .

$u = \frac{3 \pm \sqrt{15}}{- 2}$

Step 7.4

Move the negative in front of the fraction.

$u = - \frac{3 \pm \sqrt{15}}{2}$

Step 8

The final answer is the combination of both solutions.

$u = - \frac{3 + \sqrt{15}}{2}, - \frac{3 - \sqrt{15}}{2}$

Step 9

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

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

Step 10

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

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

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

Step 11

Solve for $x$ in $\cos (x) = - \frac{3 + \sqrt{15}}{2}$ .

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Step 11.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{3 + \sqrt{15}}{2}$ does not fall in this range, there is no solution.

No solution

Step 12

Solve for $x$ in $\cos (x) = - \frac{3 - \sqrt{15}}{2}$ .

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Step 12.1

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

$x = \arccos (- \frac{3 - \sqrt{15}}{2})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arccos (- \frac{3 - \sqrt{15}}{2})$ .

$x = 1.11910076$

Step 12.3

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

$x = 2 (3.14159265) - 1.11910076$

Step 12.4

Solve for $x$ .

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Step 12.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.11910076$

Step 12.4.2

Simplify $2 (3.14159265) - 1.11910076$ .

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Step 12.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.11910076$

Step 12.4.2.2

Subtract $1.11910076$ from $6.2831853$ .

$x = 5.16408454$

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.

$x = 1.11910076 + 2 π n, 5.16408454 + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = 1.11910076 + 2 π n, 5.16408454 + 2 π n$ , for any integer $n$