Trigonometry Examples

$(\cot (x) + 1) (\sin (x) + 1) = 0$

Step 1

Simplify $(\cot (x) + 1) (\sin (x) + 1)$ .

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Step 1.1

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

$(\frac{\cos (x)}{\sin (x)} + 1) (\sin (x) + 1) = 0$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$\frac{\cos (x)}{\sin (x)} (\sin (x) + 1) + 1 (\sin (x) + 1) = 0$

Step 1.2.2

Apply the distributive property.

$\frac{\cos (x)}{\sin (x)} \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 + 1 (\sin (x) + 1) = 0$

Step 1.2.3

Apply the distributive property.

$\frac{\cos (x)}{\sin (x)} \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 + 1 \sin (x) + 1 \cdot 1 = 0$

Step 1.3

Simplify each term.

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Step 1.3.1

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

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Step 1.3.1.1

Cancel the common factor.

$\frac{\cos (x)}{\sin (x)} \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 + 1 \sin (x) + 1 \cdot 1 = 0$

Step 1.3.1.2

Rewrite the expression.

$\cos (x) + \frac{\cos (x)}{\sin (x)} \cdot 1 + 1 \sin (x) + 1 \cdot 1 = 0$

Step 1.3.2

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

$\cos (x) + \frac{\cos (x)}{\sin (x)} + 1 \sin (x) + 1 \cdot 1 = 0$

Step 1.3.3

Multiply $\sin (x)$ by $1$ .

$\cos (x) + \frac{\cos (x)}{\sin (x)} + \sin (x) + 1 \cdot 1 = 0$

Step 1.3.4

Multiply $1$ by $1$ .

$\cos (x) + \frac{\cos (x)}{\sin (x)} + \sin (x) + 1 = 0$

Step 1.4

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

$\cos (x) + \cot (x) + \sin (x) + 1 = 0$

Step 2

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

$x = \frac{3 π}{4} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 3