Problem 87 - A Graphical Approach to Precalculus with Limits

Question

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\cos \theta>0, \sin \theta>0$$

Step-by-Step Solution

Answer

The angle \(\theta\) is in Quadrant I.

Step 1: Understand the Conditions

The problem provides two conditions: \(\cos \theta > 0\) and \(\sin \theta > 0\). This means that both the cosine and sine of the angle \(\theta\) are positive.

Step 2: Analyze Quadrants

Recall that the unit circle is divided into four quadrants based on the signs of \(\sin \theta\) and \(\cos \theta\): - Quadrant I: \(\sin \theta > 0\), \(\cos \theta > 0\) - Quadrant II: \(\sin \theta > 0\), \(\cos \theta < 0\) - Quadrant III: \(\sin \theta < 0\), \(\cos \theta < 0\) - Quadrant IV: \(\sin \theta < 0\), \(\cos \theta > 0\).

Step 3: Identify the Appropriate Quadrant

Based on the conditions \(\cos \theta > 0\) and \(\sin \theta > 0\), we find that this is true only in Quadrant I. In this quadrant, both sine and cosine values are positive.

Key Concepts

Unit Circle

The unit circle is an essential tool in trigonometry that helps us understand the relationships between angles and trigonometric functions. Picture a circle in the coordinate plane with its center at the origin and a radius of one unit. This is the unit circle, and it forms the basis for defining sine and cosine functions.

In the unit circle:

• Each point on the circle can be described by the coordinates \( (x, y) \).
• The angle \( \theta \) is measured from the positive x-axis.
• The x-coordinate is equivalent to \( \cos \theta \), and the y-coordinate is equivalent to \( \sin \theta \).

Additionally, the unit circle is divided into four quadrants, each with unique sign rules for sine and cosine. These quadrants help determine the positivity or negativity of the trigonometric functions.

Knowing which quadrant an angle lies in can greatly simplify solving trigonometric problems. For instance, the original problem determines the angle is in the first quadrant due to the positivity of both sine and cosine values.

Sine Function

The sine function is closely tied to the y-coordinate of a point on the unit circle. When we have an angle \( \theta \), the sine of \( \theta \) is the y-value of where the terminal side of the angle intersects with the unit circle.

Here are some key points about sine that are useful to remember:

• In Quadrant I, \( \sin \theta \) is positive because the point on the unit circle is above the x-axis.
• In Quadrant II, \( \sin \theta \) is still positive, but the x-coordinate (\( \cos \theta \)) changes to negative.
• In Quadrants III and IV, \( \sin \theta \) becomes negative.

The sine function is periodic and has a range of [-1, 1]. This makes sine a vital tool in understanding periodic phenomena.

In the provided example exercise, knowing \( \sin \theta > 0 \) immediately excludes angles in Quadrants III and IV.

Cosine Function

Cosine is another fundamental trigonometric function, represented by the x-coordinate of a point on the unit circle. For an angle \( \theta \), \( \cos \theta \) tells us how far left or right the point is from the origin.

Important characteristics of the cosine function include:

• \( \cos \theta > 0 \) in Quadrants I and IV, where the point lies on the right side of the y-axis.
• \( \cos \theta < 0 \) in Quadrants II and III, where the point is on the left side.

Cosine also has a range of [-1, 1] and, like sine, is periodic. These properties make the cosine function very useful in solving and modeling problems involving cycles.

For the exercise, the condition \( \cos \theta > 0 \) ensures that \( \theta \) cannot be in Quadrants II or III. Coupled with the positivity of the sine function, this confirms that the angle must reside in Quadrant I.