Precalculus Examples

$(\cot (x) - \sqrt{3}) (\sqrt{2} \sin (x) + 1) = 0$

Step 1

Simplify $(\cot (x) - \sqrt{3}) (\sqrt{2} \sin (x) + 1)$ .

Tap for more steps...

Step 1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$(\frac{\cos (x)}{\sin (x)} - \sqrt{3}) (\sqrt{2} \sin (x) + 1) = 0$

Step 1.2

Expand $(\frac{\cos (x)}{\sin (x)} - \sqrt{3}) (\sqrt{2} \sin (x) + 1)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\frac{\cos (x)}{\sin (x)} (\sqrt{2} \sin (x) + 1) - \sqrt{3} (\sqrt{2} \sin (x) + 1) = 0$

Step 1.2.2

Apply the distributive property.

$\frac{\cos (x)}{\sin (x)} (\sqrt{2} \sin (x)) + \frac{\cos (x)}{\sin (x)} \cdot 1 - \sqrt{3} (\sqrt{2} \sin (x) + 1) = 0$

Step 1.2.3

Apply the distributive property.

$\frac{\cos (x)}{\sin (x)} (\sqrt{2} \sin (x)) + \frac{\cos (x)}{\sin (x)} \cdot 1 - \sqrt{3} (\sqrt{2} \sin (x)) - \sqrt{3} \cdot 1 = 0$

Step 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

Rewrite using the commutative property of multiplication.

$\sqrt{2} \frac{\cos (x)}{\sin (x)} \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 - \sqrt{3} (\sqrt{2} \sin (x)) - \sqrt{3} \cdot 1 = 0$

Step 1.3.2

Combine $\sqrt{2}$ and $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\sqrt{2} \cos (x)}{\sin (x)} \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 - \sqrt{3} (\sqrt{2} \sin (x)) - \sqrt{3} \cdot 1 = 0$

Step 1.3.3

Cancel the common factor of $\sin (x)$ .

Tap for more steps...

Step 1.3.3.1

Cancel the common factor.

$\frac{\sqrt{2} \cos (x)}{\sin (x)} \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 - \sqrt{3} (\sqrt{2} \sin (x)) - \sqrt{3} \cdot 1 = 0$

Step 1.3.3.2

Rewrite the expression.

$\sqrt{2} \cos (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 - \sqrt{3} (\sqrt{2} \sin (x)) - \sqrt{3} \cdot 1 = 0$

Step 1.3.4

Multiply $\frac{\cos (x)}{\sin (x)}$ by $1$ .

$\sqrt{2} \cos (x) + \frac{\cos (x)}{\sin (x)} - \sqrt{3} (\sqrt{2} \sin (x)) - \sqrt{3} \cdot 1 = 0$

Step 1.3.5

Multiply $- \sqrt{3} (\sqrt{2} \sin (x))$ .

Tap for more steps...

Step 1.3.5.1

Combine using the product rule for radicals.

$\sqrt{2} \cos (x) + \frac{\cos (x)}{\sin (x)} - \sqrt{2 \cdot 3} \sin (x) - \sqrt{3} \cdot 1 = 0$

Step 1.3.5.2

Multiply $2$ by $3$ .

$\sqrt{2} \cos (x) + \frac{\cos (x)}{\sin (x)} - \sqrt{6} \sin (x) - \sqrt{3} \cdot 1 = 0$

Step 1.3.6

Multiply $- 1$ by $1$ .

$\sqrt{2} \cos (x) + \frac{\cos (x)}{\sin (x)} - \sqrt{6} \sin (x) - \sqrt{3} = 0$

Step 1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\sqrt{2} \cos (x) + \cot (x) - \sqrt{6} \sin (x) - \sqrt{3} = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{6} + 2 π n, \frac{7 π}{6} + 2 π n, \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 3