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 the signs of tangent and cosine functions in each quadrant: - Quadrant I: tan > 0, cos > 0 - Quadrant II: tan < 0, cos < 0 - Quadrant III: tan > 0, cos < 0 - Quadrant IV: tan < 0, cos > 0
2. Analyze the condition tan \(\theta\) < 0. This means the angle \(\theta\) must lie in a quadrant where tangent is negative, which are Quadrants II and IV.
3. Analyze the condition cos \(\theta\) < 0. This means the angle \(\theta\) must lie in a quadrant where cosine is negative, which are Quadrants II and III.
4. Find the intersection of the two sets of quadrants from the above conditions: - tan \(\theta\) < 0 gives Quadrants II and IV - cos \(\theta\) < 0 gives Quadrants II and III The common quadrant is Quadrant II.
5. Conclude that the angle \(\theta\) satisfying both tan \(\theta\) < 0 and cos \(\theta\) < 0 lies in Quadrant II.

Key Concepts

Signs of Trigonometric Functions in Quadrants

The signs of sine, cosine, and tangent functions vary depending on the quadrant of the angle. For example, cosine is positive in the first and fourth quadrants, while tangent is positive in the first and third quadrants. Understanding these sign patterns helps determine the possible quadrants for a given angle.

Relationship Between Tangent and Sine/Cosine

Tangent of an angle is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). The sign of tangent depends on the signs of sine and cosine, so knowing the sign of tangent and cosine allows inference about the sine sign and thus the quadrant.

Quadrant Identification Using Inequalities

Given inequalities like tan θ < 0 and cos θ < 0, one can use the known sign patterns of trig functions in each quadrant to identify which quadrants satisfy both conditions simultaneously. This method involves matching the signs to the correct quadrant(s).