Trigonometry Examples

$4 \sin^{2} (x) = 2 \cos (x) + 1$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $2 \cos (x)$ from both sides of the equation.

$4 \sin^{2} (x) - 2 \cos (x) = 1$

Step 1.2

Subtract $1$ from both sides of the equation.

$4 \sin^{2} (x) - 2 \cos (x) - 1 = 0$

Step 2

Replace the $4 \sin^{2} (x)$ with $4 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$4 (1 - \cos^{2} (x)) - 2 \cos (x) - 1 = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$4 \cdot 1 + 4 (- \cos^{2} (x)) - 2 \cos (x) - 1 = 0$

Step 3.2

Multiply $4$ by $1$ .

$4 + 4 (- \cos^{2} (x)) - 2 \cos (x) - 1 = 0$

Step 3.3

Multiply $- 1$ by $4$ .

$4 - 4 \cos^{2} (x) - 2 \cos (x) - 1 = 0$

Step 4

Subtract $1$ from $4$ .

$- 4 \cos^{2} (x) - 2 \cos (x) + 3 = 0$

Step 5

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

$- 4 {(u)}^{2} - 2 u + 3 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = - 4$ , $b = - 2$ , and $c = 3$ into the quadratic formula and solve for $u$ .

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

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

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

Step 8.1.2

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

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Step 8.1.2.1

Multiply $- 4$ by $- 4$ .

$u = \frac{2 \pm \sqrt{4 + 16 \cdot 3}}{2 \cdot - 4}$

Step 8.1.2.2

Multiply $16$ by $3$ .

$u = \frac{2 \pm \sqrt{4 + 48}}{2 \cdot - 4}$

Step 8.1.3

Add $4$ and $48$ .

$u = \frac{2 \pm \sqrt{52}}{2 \cdot - 4}$

Step 8.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 8.1.4.1

Factor $4$ out of $52$ .

$u = \frac{2 \pm \sqrt{4 (13)}}{2 \cdot - 4}$

Step 8.1.4.2

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

$u = \frac{2 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot - 4}$

Step 8.1.5

Pull terms out from under the radical.

$u = \frac{2 \pm 2 \sqrt{13}}{2 \cdot - 4}$

Step 8.2

Multiply $2$ by $- 4$ .

$u = \frac{2 \pm 2 \sqrt{13}}{- 8}$

Step 8.3

Simplify $\frac{2 \pm 2 \sqrt{13}}{- 8}$ .

$u = \frac{1 \pm \sqrt{13}}{- 4}$

Step 8.4

Move the negative in front of the fraction.

$u = - \frac{1 \pm \sqrt{13}}{4}$

Step 9

The final answer is the combination of both solutions.

$u = - \frac{1 + \sqrt{13}}{4}, - \frac{1 - \sqrt{13}}{4}$

Step 10

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

$\cos (x) = - \frac{1 + \sqrt{13}}{4}, - \frac{1 - \sqrt{13}}{4}$

Step 11

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

$\cos (x) = - \frac{1 + \sqrt{13}}{4}$

$\cos (x) = - \frac{1 - \sqrt{13}}{4}$

Step 12

Solve for $x$ in $\cos (x) = - \frac{1 + \sqrt{13}}{4}$ .

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Step 12.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{1 + \sqrt{13}}{4}$ does not fall in this range, there is no solution.

No solution

Step 13

Solve for $x$ in $\cos (x) = - \frac{1 - \sqrt{13}}{4}$ .

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Step 13.1

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

$x = \arccos (- \frac{1 - \sqrt{13}}{4})$

Step 13.2

Simplify the right side.

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Step 13.2.1

Evaluate $\arccos (- \frac{1 - \sqrt{13}}{4})$ .

$x = 0.86138422$

Step 13.3

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.

$x = 2 (3.14159265) - 0.86138422$

Step 13.4

Solve for $x$ .

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Step 13.4.1

Remove parentheses.

$x = 2 (3.14159265) - 0.86138422$

Step 13.4.2

Simplify $2 (3.14159265) - 0.86138422$ .

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Step 13.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.86138422$

Step 13.4.2.2

Subtract $0.86138422$ from $6.2831853$ .

$x = 5.42180108$

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.

$x = 0.86138422 + 2 π n, 5.42180108 + 2 π n$ , for any integer $n$

Step 14

List all of the solutions.

$x = 0.86138422 + 2 π n, 5.42180108 + 2 π n$ , for any integer $n$