Trigonometry Examples

$\tan (15 π - 2 t)$

Step 1

Use the difference formula for tangent to simplify the expression. The formula states that $\tan (A - B) = \frac{\tan (A) - \tan (B)}{1 + \tan (A) \tan (B)}$ .

$\frac{\tan (15 π) - \tan (2 t)}{1 + \tan (15 π) \tan (2 t)}$

Step 2

Remove parentheses.

$\frac{\tan (15 π) - \tan (2 t)}{1 + \tan (15 π) \tan (2 t)}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $- 1$ out of $\tan (15 π) - \tan (2 t)$ .

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Step 3.1.1

Factor $- 1$ out of $\tan (15 π)$ .

$\frac{- 1 (- \tan (15 π)) - \tan (2 t)}{1 + \tan (15 π) \tan (2 t)}$

Step 3.1.2

Factor $- 1$ out of $- \tan (2 t)$ .

$\frac{- 1 (- \tan (15 π)) - (\tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- \tan (15 π)) - (\tan (2 t))$ .

$\frac{- 1 (- \tan (15 π) + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.1.4

Rewrite $- 1 (- \tan (15 π) + \tan (2 t))$ as $- (- \tan (15 π) + \tan (2 t))$ .

$\frac{- (- \tan (15 π) + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.2

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

$\frac{- (- \tan (π) + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- (- - \tan (0) + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.4

The exact value of $\tan (0)$ is $0$ .

$\frac{- (- - 0 + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.5

Multiply $- - 0$ .

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Step 3.5.1

Multiply $- 1$ by $0$ .

$\frac{- (- 0 + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.5.2

Multiply $- 1$ by $0$ .

$\frac{- (0 + \tan (2 t))}{1 + \tan (15 π) \tan (2 t)}$

Step 3.6

Add $0$ and $\tan (2 t)$ .

$\frac{- \tan (2 t)}{1 + \tan (15 π) \tan (2 t)}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{- \tan (2 t)}{1 + \tan (π) \tan (2 t)}$

Step 4.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{- \tan (2 t)}{1 - \tan (0) \tan (2 t)}$

Step 4.3

The exact value of $\tan (0)$ is $0$ .

$\frac{- \tan (2 t)}{1 - 0 \tan (2 t)}$

Step 4.4

Multiply $- 1$ by $0$ .

$\frac{- \tan (2 t)}{1 + 0 \tan (2 t)}$

Step 4.5

Multiply $0$ by $\tan (2 t)$ .

$\frac{- \tan (2 t)}{1 + 0}$

Step 4.6

Add $1$ and $0$ .

$\frac{- \tan (2 t)}{1}$

Step 5

Divide $- \tan (2 t)$ by $1$ .

$- \tan (2 t)$