Trigonometry Examples

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

Step 1

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

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Step 1.1

Add $\sin^{2} (x)$ to both sides of the equation.

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

Step 1.2

Subtract $4$ from both sides of the equation.

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

Step 2

Replace $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Apply pythagorean identity.

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

Step 3.2

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

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

Step 3.3

Subtract $4$ from $1$ .

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

Step 3.4

Add $3$ to both sides of the equation.

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

Step 3.5

Divide each term in $- \cos^{2} (x) = 3$ by $- 1$ and simplify.

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Step 3.5.1

Divide each term in $- \cos^{2} (x) = 3$ by $- 1$ .

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.2.2

Divide $\cos^{2} (x)$ by $1$ .

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $3$ by $- 1$ .

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

Step 3.6

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 3.7

Simplify $\pm \sqrt{- 3}$ .

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Step 3.7.1

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 3.7.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$\cos (x) = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 3.7.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\cos (x) = \pm i \sqrt{3}$

Step 3.8

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.8.1

First, use the positive value of the $\pm$ to find the first solution.

$\cos (x) = i \sqrt{3}$

Step 3.8.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.8.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 3.9

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

$\cos (x) = i \sqrt{3}$

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

Step 3.10

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

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Step 3.10.1

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

$x = \arccos (i \sqrt{3})$

Step 3.10.2

The inverse cosine of $\arccos (i \sqrt{3})$ is undefined.

Undefined

Step 3.11

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

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Step 3.11.1

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

$x = \arccos (- i \sqrt{3})$

Step 3.11.2

The inverse cosine of $\arccos (- i \sqrt{3})$ is undefined.

Undefined

Step 3.12

List all of the solutions.

No solution