Using reference angles, what is the exact value of $\tan (195°)$?

• A. $\frac{1}{\sqrt{3}}$
• B. $-\sqrt{3}$
• C. $\sqrt{3}$
• D. $-\frac{1}{\sqrt{3}}$

Answer: (A)

Verified step by step guidance

1. Identify the quadrant in which the angle 195° lies. Since 195° is between 180° and 270°, it is in the third quadrant.
2. Find the reference angle for 195°. The reference angle \( \theta_r \) is the acute angle between 195° and the nearest x-axis, calculated as \( \theta_r = 195° - 180° \).
3. Recall the sign of the tangent function in the third quadrant. Tangent is positive in the third quadrant.
4. Use the reference angle to find the exact value of \( \tan(195°) \). Since \( \tan(\theta) = \tan(\theta_r) \) in the third quadrant, express \( \tan(195°) \) in terms of \( \tan(\theta_r) \).
5. Evaluate \( \tan(\theta_r) \) using known special angles (like 15°, 30°, 45°, 60°, 75°) or by using the tangent subtraction formula if needed, then apply the sign determined in step 3.