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

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Recall the signs of the trigonometric functions in each quadrant: In Quadrant I, all functions are positive; in Quadrant II, sine is positive while cosine and tangent are negative; in Quadrant III, tangent is positive while sine and cosine are negative; in Quadrant IV, cosine is positive while sine and tangent are negative.

Analyze the given conditions: \(\tan \theta > 0\) means tangent is positive, and \(\cos \theta < 0\) means cosine is negative.

From the sign chart, identify the quadrants where tangent is positive: Quadrant I and Quadrant III.

Identify the quadrants where cosine is negative: Quadrant II and Quadrant III.

Find the quadrant common to both conditions (tangent positive and cosine negative), which is Quadrant III.

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Sign of Trigonometric Functions in Quadrants

The signs of sine, cosine, and tangent functions vary depending on the quadrant in which the angle lies. Knowing these sign patterns helps determine the quadrant based on given inequalities for trigonometric values.

Relationship Between Tangent, Sine, and Cosine

Tangent of an angle is the ratio of sine to cosine (tan θ = sin θ / cos θ). Understanding this relationship allows us to infer the signs of sine and cosine from the sign of tangent and vice versa.

Quadrant Identification Using Inequalities

By analyzing the given inequalities (tan θ > 0 and cos θ < 0), we can identify the quadrant where both conditions hold true. This involves matching the sign conditions to the known sign patterns of trig functions in each quadrant.

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