Trigonometry Examples

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

Step 1

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

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

Step 2

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

$2 \cos (x) + 3 - (1 - \cos^{2} (x)) = 0$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

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

$2 (u) + 3 - (1 - {(u)}^{2}) = 0$

Step 3.2

Simplify $2 u + 3 - (1 - u^{2})$ .

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$2 u + 3 - 1 \cdot 1 - - u^{2} = 0$

Step 3.2.1.2

Multiply $- 1$ by $1$ .

$2 u + 3 - 1 - - u^{2} = 0$

Step 3.2.1.3

Multiply $- - u^{2}$ .

Tap for more steps...

Step 3.2.1.3.1

Multiply $- 1$ by $- 1$ .

$2 u + 3 - 1 + 1 u^{2} = 0$

Step 3.2.1.3.2

Multiply $u^{2}$ by $1$ .

$2 u + 3 - 1 + u^{2} = 0$

Step 3.2.2

Subtract $1$ from $3$ .

$2 u + 2 + u^{2} = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = 1$ , $b = 2$ , and $c = 2$ into the quadratic formula and solve for $u$ .

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

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Simplify the numerator.

Tap for more steps...

Step 3.5.1.1

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

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

Step 3.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.5.1.2.2

Multiply $- 4$ by $2$ .

$u = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 3.5.1.3

Subtract $8$ from $4$ .

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

Step 3.5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 3.5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 3.5.1.6

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

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

Step 3.5.1.7

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

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

Step 3.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{- 2 \pm i \cdot 2}{2 \cdot 1}$

Step 3.5.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i}{2 \cdot 1}$

Step 3.5.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i}{2}$

Step 3.5.3

Simplify $\frac{- 2 \pm 2 i}{2}$ .

$u = - 1 \pm i$

Step 3.6

The final answer is the combination of both solutions.

$u = - 1 + i, - 1 - i$

Step 3.7

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

$\cos (x) = - 1 + i, - 1 - i$

Step 3.8

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

$\cos (x) = - 1 + i$

$\cos (x) = - 1 - i$

Step 3.9

Solve for $x$ in $\cos (x) = - 1 + i$ .

Tap for more steps...

Step 3.9.1

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

$x = \arccos (- 1 + i)$

Step 3.9.2

The inverse cosine of $\arccos (- 1 + i)$ is undefined.

Undefined

Step 3.10

Solve for $x$ in $\cos (x) = - 1 - i$ .

Tap for more steps...

Step 3.10.1

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

$x = \arccos (- 1 - i)$

Step 3.10.2

The inverse cosine of $\arccos (- 1 - i)$ is undefined.

Undefined

Step 3.11

List all of the solutions.

No solution