Trigonometry Examples

$(\tan (θ) - 1) (\cos (θ) + 1) = 0$

Step 1

Simplify $(\tan (θ) - 1) (\cos (θ) + 1)$ .

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Step 1.1

Rewrite $\tan (θ)$ in terms of sines and cosines.

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify each term.

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Step 1.3.1

Cancel the common factor of $\cos (θ)$ .

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Step 1.3.1.1

Cancel the common factor.

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

Step 1.3.1.2

Rewrite the expression.

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

Step 1.3.2

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

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

Step 1.3.3

Rewrite $- 1 \cos (θ)$ as $- \cos (θ)$ .

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

Step 1.3.4

Multiply $- 1$ by $1$ .

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

Step 1.4

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$\sin (θ) + \tan (θ) - \cos (θ) - 1 = 0$

Step 2

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

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

Step 3