Precalculus Examples

$\tan (\frac{π}{6} - \frac{2 π}{3})$

Step 1

Rewrite $\tan (\frac{π}{6} - \frac{2 π}{3})$ in terms of sines and cosines.

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π}{6} - \frac{2 π}{3})}$

Step 2

To write $- \frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π}{6} - \frac{2 π}{3} \cdot \frac{2}{2})}$

Step 3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2 π}{3}$ by $\frac{2}{2}$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π}{6} - \frac{2 π \cdot 2}{3 \cdot 2})}$

Step 3.2

Multiply $3$ by $2$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π}{6} - \frac{2 π \cdot 2}{6})}$

Step 4

Combine the numerators over the common denominator.

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π - 2 π \cdot 2}{6})}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $- 2$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π - 4 π}{6})}$

Step 5.2

Subtract $4 π$ from $π$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{- 3 π}{6})}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $- 3$ and $6$ .

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Step 6.1.1

Factor $3$ out of $- 3 π$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{3 (- π)}{6})}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $3$ out of $6$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{3 (- π)}{3 (2)})}$

Step 6.1.2.2

Cancel the common factor.

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{3 (- π)}{3 \cdot 2})}$

Step 6.1.2.3

Rewrite the expression.

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{- π}{2})}$

Step 6.2

Move the negative in front of the fraction.

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (- \frac{π}{2})}$

Step 7

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{3 π}{2})}$

Step 8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{\cos (\frac{π}{2})}$

Step 9

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{\sin (\frac{π}{6} - \frac{2 π}{3})}{0}$

Step 10

The expression contains a division by $0$ . The expression is undefined.

Undefined