Roots and Tangent Formula Given that and are roots of , find .

Trigonometry Solution – Problem 3: find

Roots and Tangent Formula

Given that $\tan\alpha$ and $\tan\beta$ are roots of $x^2 - 7x + 13 = 0$ , find $\tan(\alpha+\beta)$ .

By Vieta's formulas:

$\tan\alpha + \tan\beta = 7, \quad \tan\alpha\cdot\tan\beta = 13$

Using the tangent sum formula:

$\tan(\alpha+\beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\cdot\tan\beta} = \frac{7}{1-13} = -\frac{7}{12}$

$-\frac{7}{12}$

Correct identity setup (2 pts): choose an appropriate sum/difference, double-angle, or auxiliary-angle idea and set up the key equation(s).
Correct algebra / trig simplification (2 pts): transform expressions without sign mistakes.
Solve for target quantity (2 pts): isolate the requested value and handle any constraints if needed.
Final answer (1 pt): clearly state the result in the required form.