Problem 18 - Precalculus

Question

Indicate the quadrant in which the terminal side of \(\theta\) must lie in order for the information to be true. \(\tan \theta\) is positive and \(\cos \theta\) is negative.

Step-by-Step Solution

Answer

The terminal side of \(\theta\) lies in the third quadrant.

Step 1: Understanding Tangent and Cosine Signs in the Quadrants

Recall that the tangent function \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) is positive when both sine and cosine have the same sign (either both positive or both negative). Cosine is the x-coordinate on the unit circle and is negative in the second and third quadrants.

Step 2: Identifying the Tangent Conditions

Since \(\tan \theta\) is positive, and \(\cos \theta < 0\), sine must also be negative for tangent to remain positive in these specific quadrants.

Step 3: Checking Conditions in Each Quadrant

- **Quadrant I**: \(\tan \theta\) is positive but \(\cos \theta\) is positive, so not suitable.- **Quadrant II**: \(\cos \theta\) is negative, but \(\tan \theta\) is negative, so not suitable.- **Quadrant III**: \(\tan \theta\) is positive (both sine and cosine are negative), and \(\cos \theta\) is negative, matching both conditions.- **Quadrant IV**: \(\tan \theta\) is negative, so not suitable.

Step 4: Conclusion

From the examination of all quadrants, the third quadrant satisfies both conditions: \(\tan \theta\) is positive and \(\cos \theta\) is negative.

Key Concepts

Unit Circle

The Unit Circle is a crucial tool in trigonometry that helps us understand the behavior of trigonometric functions such as sine, cosine, and tangent. It is a circle with a radius of one, centered at the origin (0, 0) on the coordinate plane.
The circle allows us to define the trigonometric functions for angles over an infinite domain. The beauty of the Unit Circle lies in its ability to represent angles and trigonometric values in terms of coordinates. For any angle \( \theta \), the coordinates of the corresponding point on the Unit Circle are \((\cos \theta, \sin \theta)\).
This means the x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine. Understanding how each coordinate changes as you move around the circle is key to mastering trigonometric concepts.

Quadrants

The coordinate plane is divided into four sections, known as quadrants. Each quadrant helps us determine the sign of trigonometric functions. In the context of a Unit Circle:

• **Quadrant I**: Both sine and cosine are positive.
• **Quadrant II**: Sine is positive, and cosine is negative.
• **Quadrant III**: Both sine and cosine are negative.
• **Quadrant IV**: Sine is negative, and cosine is positive.

This division is essential for understanding where each trigonometric function is positive or negative, allowing us to analyze angles based on their positions on the Unit Circle. Knowing the signs of sine and cosine in each quadrant simplifies determining the sign of the tangent function, which depends on both sine and cosine.

Tangent

The Tangent function, often expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), is one of the fundamental trigonometric functions. On the Unit Circle, tangent reflects the relationship between sine and cosine at a given angle.
For tangent to be positive, both sine and cosine must share the same sign. This means that tangent is positive in Quadrant I and Quadrant III. On the contrary, it is negative in Quadrant II and Quadrant IV.
This also affects how angles are interpreted in relation to tangent. Being able to determine where the tangent is positive or negative is useful for solving trigonometric equations and understanding function behavior across different quadrants.

Cosine

The cosine function, represented as \( \cos \theta \), is associated with the x-coordinate of points on the Unit Circle. Since it relates to the horizontal distance from the origin to the point where the terminal side of \( \theta \) intersects the circle, cosine provides information about this distance.
As such, its sign changes as one moves through the quadrants:

• In **Quadrant I**, cosine is positive.
• In **Quadrant II**, cosine becomes negative.
• In **Quadrant III**, it remains negative.
• In **Quadrant IV**, cosine is positive again.

When analyzing the conditions \( \cos \theta < 0 \), you'll focus on Quadrant II and Quadrant III. However, to satisfy the problem condition where \( \tan \theta \) is also positive, Quadrant III becomes the ideal zone.