Trigonometry Examples

$\tan (u - 5 π)$

Step 1

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

$\frac{\tan (u) - \tan (5 π)}{1 + \tan (u) \tan (5 π)}$

Step 2

Remove parentheses.

$\frac{\tan (u) - \tan (5 π)}{1 + \tan (u) \tan (5 π)}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{\tan (u) - \tan (π)}{1 + \tan (u) \tan (5 π)}$

Step 3.2

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.

$\frac{\tan (u) - - \tan (0)}{1 + \tan (u) \tan (5 π)}$

Step 3.3

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

$\frac{\tan (u) - - 0}{1 + \tan (u) \tan (5 π)}$

Step 3.4

Multiply $- - 0$ .

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Step 3.4.1

Multiply $- 1$ by $0$ .

$\frac{\tan (u) - 0}{1 + \tan (u) \tan (5 π)}$

Step 3.4.2

Multiply $- 1$ by $0$ .

$\frac{\tan (u) + 0}{1 + \tan (u) \tan (5 π)}$

Step 3.5

Add $\tan (u)$ and $0$ .

$\frac{\tan (u)}{1 + \tan (u) \tan (5 π)}$

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 $2 π$ .

$\frac{\tan (u)}{1 + \tan (u) \tan (π)}$

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.

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

Step 4.3

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

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

Step 4.4

Multiply $- 1$ by $0$ .

$\frac{\tan (u)}{1 + \tan (u) \cdot 0}$

Step 4.5

Multiply $\tan (u)$ by $0$ .

$\frac{\tan (u)}{1 + 0}$

Step 4.6

Add $1$ and $0$ .

$\frac{\tan (u)}{1}$

Step 5

Divide $\tan (u)$ by $1$ .

$\tan (u)$