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

Verified step by step guidance

1. Recall the definitions and signs of the trigonometric functions in each quadrant. The tangent function \(\tan \theta\) is positive in Quadrants I and III, and negative in Quadrants II and IV. The cotangent function \(\cot \theta\) has the same sign as tangent because \(\cot \theta = \frac{1}{\tan \theta}\), so it is also positive in Quadrants I and III, and negative in Quadrants II and IV.
2. Analyze the given inequalities: \(\tan \theta < 0\) means \(\theta\) lies in a quadrant where tangent is negative, which are Quadrants II and IV.
3. Similarly, \(\cot \theta < 0\) means cotangent is negative, so \(\theta\) must be in a quadrant where cotangent is negative, which are also Quadrants II and IV.
4. Since both \(\tan \theta\) and \(\cot \theta\) are negative, the angle \(\theta\) must be in the intersection of the quadrants where both are negative, which are Quadrants II and IV.
5. Therefore, the possible quadrants for \(\theta\) are Quadrant II and Quadrant IV.

Key Concepts

Signs of Trigonometric Functions in Quadrants

The signs of sine, cosine, tangent, and cotangent vary depending on the quadrant of the angle. Tangent and cotangent are positive or negative based on the signs of sine and cosine, since tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Understanding which quadrants yield positive or negative values for these functions is essential to determine the angle's location.

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal functions: tan θ = 1 / cot θ and cot θ = 1 / tan θ. Their signs are linked but can differ depending on the quadrant. Recognizing that both tan θ and cot θ depend on sine and cosine helps analyze their signs simultaneously to narrow down possible quadrants.

Quadrant Identification Using Inequalities

Given inequalities like tan θ < 0 and cot θ < 0, one must use the sign rules of trigonometric functions to identify which quadrants satisfy these conditions. Since tangent and cotangent change signs in specific quadrants, analyzing these inequalities helps pinpoint the possible quadrants where the angle θ lies.