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 that the tangent function, \( \tan \theta \), is negative in the second and fourth quadrants.
2. Recall that the cotangent function, \( \cot \theta \), is the reciprocal of the tangent function, \( \cot \theta = \frac{1}{\tan \theta} \), and is also negative in the second and fourth quadrants.
3. Since both \( \tan \theta < 0 \) and \( \cot \theta < 0 \), we need to find the quadrants where both conditions are satisfied.
4. In the second quadrant, \( \tan \theta < 0 \) and \( \cot \theta < 0 \) because sine is positive and cosine is negative.
5. In the fourth quadrant, \( \tan \theta < 0 \) and \( \cot \theta < 0 \) because sine is negative and cosine is positive.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as tangent (tan) and cotangent (cot), relate the angles of a triangle to the ratios of its sides. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side, while the cotangent is the reciprocal of the tangent. Understanding these functions is essential for determining the signs of tan θ and cot θ in different quadrants.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, each defined by the signs of the x (cosine) and y (sine) coordinates. In Quadrant I, both sine and cosine are positive; in Quadrant II, sine is positive and cosine is negative; in Quadrant III, both are negative; and in Quadrant IV, sine is negative and cosine is positive. Identifying the correct quadrant is crucial for determining where the angle θ lies based on the signs of its trigonometric functions.

Sign of Trigonometric Functions

The signs of trigonometric functions vary depending on the quadrant in which the angle lies. For the conditions tan θ < 0 and cot θ < 0, we need to analyze where these functions are negative. Since tangent is negative in Quadrants II and IV, and cotangent is negative in Quadrants I and III, the angle θ must be in Quadrant II or IV to satisfy both conditions.