Trigonometry Examples

$\cos^{2} (x) + 6 \cos (x) + 4 = 0$

Step 1

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

${(u)}^{2} + 6 (u) + 4 = 0$

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 $a = 1$ , $b = 6$ , and $c = 4$ into the quadratic formula and solve for $u$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$u = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{- 6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Step 4.1.3

Subtract $16$ from $36$ .

$u = \frac{- 6 \pm \sqrt{20}}{2 \cdot 1}$

Step 4.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 4.1.4.1

Factor $4$ out of $20$ .

$u = \frac{- 6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 4.1.4.2

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

$u = \frac{- 6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$u = \frac{- 6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$u = \frac{- 6 \pm 2 \sqrt{5}}{2}$

Step 4.3

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

$u = - 3 \pm \sqrt{5}$

Step 5

The final answer is the combination of both solutions.

$u = - 3 + \sqrt{5}, - 3 - \sqrt{5}$

Step 6

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

$\cos (x) = - 3 + \sqrt{5}, - 3 - \sqrt{5}$

Step 7

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

$\cos (x) = - 3 + \sqrt{5}$

$\cos (x) = - 3 - \sqrt{5}$

Step 8

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

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Step 8.1

Convert the right side of the equation to its decimal equivalent.

$\cos (x) = - 0.76393202$

Step 8.2

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

$x = \arccos (- 0.76393202)$

Step 8.3

Simplify the right side.

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Step 8.3.1

Evaluate $\arccos (- 0.76393202)$ .

$x = 2.44018101$

Step 8.4

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.

$x = 2 (3.14159265) - 2.44018101$

Step 8.5

Solve for $x$ .

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Step 8.5.1

Remove parentheses.

$x = 2 (3.14159265) - 2.44018101$

Step 8.5.2

Simplify $2 (3.14159265) - 2.44018101$ .

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Step 8.5.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.44018101$

Step 8.5.2.2

Subtract $2.44018101$ from $6.2831853$ .

$x = 3.84300429$

Step 8.6

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

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Step 8.6.1

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

$\frac{2 π}{| b |}$

Step 8.6.2

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

$\frac{2 π}{| 1 |}$

Step 8.6.3

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

$\frac{2 π}{1}$

Step 8.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.7

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

$x = 2.44018101 + 2 π n, 3.84300429 + 2 π n$ , for any integer $n$

Step 9

Solve for $x$ in $\cos (x) = - 3 - \sqrt{5}$ .

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Step 9.1

Convert the right side of the equation to its decimal equivalent.

$\cos (x) = - 5.23606797$

Step 9.2

The range of cosine is $- 1 \leq y \leq 1$ . Since $- 5.23606797$ does not fall in this range, there is no solution.

No solution

Step 10

List all of the solutions.

$x = 2.44018101 + 2 π n, 3.84300429 + 2 π n$ , for any integer $n$