Solving for Quadrants Where ( sin theta 0 ) and ( tan theta ) is Undefined

When dealing with trigonometric functions on the unit circle, it is essential to understand how the quadrants and angles relate to the values of sin θ and tan θ. Different texts and authors may assign different orders to the quadrants, but the unit circle provides a consistent framework for understanding these trigonometric functions.

Trigonometric Quadrants and Coordinate Axes

In a typical trigonometry class, the positive x-axis is oriented horizontally and to the right, and the positive y-axis is vertical and up. Angles are measured counter-clockwise from the positive x-axis.

The axes and origin can be placed in various configurations, but for simplicity, we can consider a semi-inclusive scheme where:

The positive x-axis is in the first quadrant. The positive y-axis is in the second quadrant. The negative x-axis is in the third quadrant. The negative y-axis is in the fourth quadrant. The origin is considered part of all four quadrants.

This scheme works well for angles running from 0 through one full rotation of the unit circle without overlap.

Trigonometric Functions and Their Values Across Quadrants

Consider the function sin θ. It starts at 0 in the first quadrant when the angle lies on the positive x-axis. As θ increases, sin θ increases to its maximum of 1 on the boundary of the second quadrant. Then, it decreases to 0 on the negative x-axis (third quadrant boundary) and attains -1 at the negative x-axis boundary (fourth quadrant). Throughout the fourth quadrant, sin θ continues to increase towards 0.

Sine is negative only in the third and fourth quadrants. Therefore, in a clockwise semi-inclusive scheme, only the fourth quadrant contains negative values of sine exclusively.

Now, consider the function tan θ. Tangent is undefined where the angle terminates on the vertical axis, which occurs in the second and fourth quadrants.

Intersection at the Third and Fourth Quadrant Boundary

At the boundary between the third and fourth quadrants, θ 3π/2. Here, sin θ -1, and tan θ is undefined. This unique intersection highlights the importance of understanding how these functions interact across the quadrants.

Conclusion

In summary, for sin θ 0 and tan θ being undefined, the only quadrant where both conditions can be met simultaneously is the fourth quadrant.

This approach to understanding trigonometric functions and quadrants provides a clear and consistent framework, ensuring that the values and undefined points of these functions are correctly identified and applied in various mathematical problems.