Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. tan θ < 0 , cos θ < 0

Verified step by step guidance

1. Recall that the tangent function, \( \tan \theta \), is negative in the second and fourth quadrants.
2. Recall that the cosine function, \( \cos \theta \), is negative in the second and third quadrants.
3. Identify the quadrant(s) where both conditions are satisfied: \( \tan \theta < 0 \) and \( \cos \theta < 0 \).
4. Determine that the second quadrant satisfies both conditions: \( \tan \theta < 0 \) and \( \cos \theta < 0 \).
5. Conclude that the angle \( \theta \) is in the second quadrant.

Key Concepts

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle θ. Quadrant I (0° to 90°) has both sine and cosine positive, Quadrant II (90° to 180°) has sine positive and cosine negative, Quadrant III (180° to 270°) has both sine and cosine negative, and Quadrant IV (270° to 360°) has sine negative and cosine positive. Understanding these quadrants is essential for determining the signs of trigonometric functions.

Signs of Trigonometric Functions

The signs of the trigonometric functions sine, cosine, and tangent vary depending on the quadrant in which the angle θ lies. Specifically, tangent is positive in Quadrants I and III, while it is negative in Quadrants II and IV. Cosine is positive in Quadrants I and IV, and negative in Quadrants II and III. This knowledge is crucial for analyzing inequalities involving these functions.

Inequalities in Trigonometry

Inequalities involving trigonometric functions, such as tan θ < 0 and cos θ < 0, require understanding the conditions under which these functions are positive or negative. For the given conditions, tan θ < 0 indicates that θ must be in Quadrants II or IV, while cos θ < 0 restricts θ to Quadrants II and III. Analyzing these inequalities helps in identifying the possible quadrants for the angle θ.