Determining the Quadrant Based on Tangent and Secant Sign

Understanding the relationship between the signs of trigonometric functions is crucial for solving problems in trigonometry. In this article, we will discuss how to determine the quadrant in which an angle terminates when given the signs of its tangent and secant. We will explore the properties of these functions and use mental associations to simplify our reasoning.

Sign of Tangent and Its Implications

The tangent of an angle, denoted as (tan(theta)), is defined as the ratio of sine to cosine ((tan(theta) frac{sin(theta)}{cos(theta)})). The sign of the tangent depends on the signs of both the sine and cosine functions. In the unit circle:

Second Quadrant: The sine is positive and the cosine is negative, making the tangent negative. Fourth Quadrant: The sine is negative and the cosine is positive, making the tangent negative.

Sign of Secant and Its Implications

The secant of an angle, denoted as (sec(theta)), is the reciprocal of the cosine ((sec(theta) frac{1}{cos(theta)})). The sign of the secant depends on the sign of the cosine.

First Quadrant: Both sine and cosine are positive, making the secant positive. Fourth Quadrant: Sine is negative and cosine is positive, making the secant positive.

Combining the Information From Tangent and Secant

Given that the tangent is negative, we know the angle must terminate in either the second or fourth quadrant. Given that the secant is positive, we know the angle must terminate in either the first or fourth quadrant. By combining these two pieces of information, we can conclude that the angle must terminate in the fourth quadrant. This is because the only quadrant where both conditions (negative tangent and positive secant) are satisfied is the fourth quadrant.

Visualizing with the Unit Circle

Mental associations can be helpful when visualizing the unit circle. Here are the key associations:

Cosine: Represents the x-coordinate on the unit circle. It is positive in the first and fourth quadrants. Sine: Represents the y-coordinate on the unit circle. It is positive in the first and second quadrants. Slope: The tangent of an angle can be thought of as the slope of the line representing the terminal side of the angle. This slope is negative in the second and fourth quadrants.

By combining these associations, we can more easily determine the quadrant in question. For example, a negative slope (tangent) and a positive x-coordinate (secant) indicate the fourth quadrant.

Sample Problem

Consider an angle where the tangent is negative and the secant is positive. Determine the quadrant in which the angle terminates.

Step 1: Identify the quadrant where the tangent is negative.

The tangent is negative in the second and fourth quadrants.

Step 2: Identify the quadrant where the secant is positive.

The secant is positive in the first and fourth quadrants.

Step 3: Combine the information to determine the quadrant in which the angle terminates.

The only quadrant that satisfies both conditions is the fourth quadrant.

Therefore, the angle terminates in the fourth quadrant.

Conclusion

By understanding the signs of the tangent and secant functions and using mental associations, we can quickly determine the quadrant in which an angle terminates when given these conditions. The quadrant in question can be found by identifying the quadrant where both the tangent is negative and the secant is positive.