Trigonometry Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

Subtract $4$ from both sides of the equation.

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

Step 2

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

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Apply pythagorean identity.

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

Step 3.2

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

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

Step 3.3

Subtract $4$ from $1$ .

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

Step 3.4

Add $3$ to both sides of the equation.

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

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.2.2

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

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

Step 3.5.3

Simplify the right side.

Tap for more steps...

Step 3.5.3.1

Divide $3$ by $- 1$ .

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

Step 3.6

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

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

Step 3.7

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

Tap for more steps...

Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.8

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

Tap for more steps...

Step 3.8.1

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

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

Step 3.8.2

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

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

Step 3.8.3

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

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

Step 3.9

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

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

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

Step 3.10

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

Tap for more steps...

Step 3.10.1

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

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

Step 3.10.2

The inverse sine of $\arcsin (i \sqrt{3})$ is undefined.

Undefined

Step 3.11

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

Tap for more steps...

Step 3.11.1

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

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

Step 3.11.2

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

Undefined

Step 3.12

List all of the solutions.

No solution