In Exercises 39–40, let θ be an angle in standard position. Name the quadrant in which θ lies.tan θ > 0 and cos θ < 0

Verified step by step guidance

1. insert step 1> Identify the trigonometric functions involved: tangent (tan) and cosine (cos).
2. insert step 2> Recall the signs of trigonometric functions in each quadrant: In Quadrant I, all are positive; in Quadrant II, sine is positive; in Quadrant III, tangent is positive; in Quadrant IV, cosine is positive.
3. insert step 3> Analyze the given conditions: tan θ > 0 indicates that θ could be in Quadrant I or III, where tangent is positive.
4. insert step 4> Consider the second condition: cos θ < 0 indicates that θ could be in Quadrant II or III, where cosine is negative.
5. insert step 5> Determine the quadrant that satisfies both conditions: Since both conditions are true in Quadrant III, θ lies in Quadrant III.

Key Concepts

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants based on the signs of the x (cosine) and y (sine) coordinates. Quadrant I has both coordinates positive, Quadrant II has a negative x and positive y, Quadrant III has both negative, and Quadrant IV has a positive x and negative y. Understanding these quadrants is essential for determining the signs of trigonometric functions in relation to the angle θ.

Trigonometric Functions and Their Signs

The tangent function (tan θ) is the ratio of sine to cosine (sin θ/cos θ). Therefore, tan θ is positive when both sine and cosine are either both positive or both negative. Conversely, cosine (cos θ) is negative in Quadrants II and III. Recognizing the relationships between these functions helps in identifying the quadrant where the angle θ lies.

Inequalities in Trigonometry

Inequalities involving trigonometric functions provide critical information about the angle's position. In this case, the condition tan θ > 0 indicates that both sine and cosine must have the same sign, while cos θ < 0 specifies that cosine is negative. Analyzing these inequalities together allows us to deduce the specific quadrant for the angle θ.