The angles that share the same tangent value as $\tan (45°)$ have terminal sides in which quadrant(s)?

• A. Quadrants II and III
• B. Quadrants I and IV
• C. Quadrants II and IV
• D. Quadrants I and III

Answer: (D)

Verified step by step guidance

1. Recall that the tangent function is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), and it is positive when both sine and cosine have the same sign.
2. Identify the sign of tangent in each quadrant: In Quadrant I, both sine and cosine are positive, so tangent is positive; in Quadrant II, sine is positive and cosine is negative, so tangent is negative; in Quadrant III, both sine and cosine are negative, so tangent is positive; in Quadrant IV, sine is negative and cosine is positive, so tangent is negative.
3. Since \(\tan 45^\circ\) is positive, angles with the same tangent value must lie in quadrants where tangent is positive, which are Quadrants I and III.
4. Understand that tangent has a period of \(180^\circ\), so angles separated by \(180^\circ\) share the same tangent value, which corresponds to angles in Quadrants I and III.
5. Therefore, the terminal sides of angles with the same tangent value as \(\tan 45^\circ\) lie in Quadrants I and III.