Trigonometry Solution – Problem 8: Angle Evaluation Evaluate

Question

Angle Evaluation

Evaluate $\frac{1 + \sqrt{3}\tan 10°}{1 + \cos 20°}$ .

Rewrite the expression:

$\frac{1 + \sqrt{3}\tan 10°}{1 + \cos 20°} = \frac{\sin 10° + \sqrt{3}\cos 10°}{\cos 10° \cdot 2\cos^2 10°}$

$= \frac{2\cos(10° - 60°)}{2\sin 10°\cos 10° \cdot \cos 10°}$

$= \frac{2\cos(-50°)}{\sin 20° \cdot \cos 10°}$

$= \frac{2\sin 40°}{\sin 20° \cdot \cos 10°}$

$= \frac{4\sin 20°\cos 20°}{\sin 20° \cdot \cos 10°} = \frac{4\cos 20°}{\cos 10°} = 2\sqrt{2}$

$2\sqrt{2}$

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.