Problem 93 - A Graphical Approach to Precalculus with Limits

Question

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\sec s<0, \csc s<0$$

Step-by-Step Solution

Answer

The point must lie in the third quadrant.

Step 1: Understand the Condition on Secant

The secant function, \( \sec s = \frac{1}{\cos s} \), is negative when the cosine is negative. The cosine is negative in the second and third quadrants. Thus, \( s \) could be in either the second or third quadrant based on this condition alone.

Step 2: Analyze the Condition on Cosecant

The cosecant function, \( \csc s = \frac{1}{\sin s} \), is negative when the sine is negative. The sine is negative in the third and fourth quadrants. Therefore, \( s \) could be in either the third or fourth quadrant based on this condition alone.

Step 3: Determine the Overlapping Quadrant

From Step 1, \( s \) could be in the second or third quadrant since \( \sec s < 0 \). From Step 2, \( s \) could be in the third or fourth quadrant since \( \csc s < 0 \). The only quadrant that satisfies both \( \sec s < 0 \) and \( \csc s < 0 \) is the third quadrant, where both cosine and sine are negative.

Step 4: Conclude the Quadrant

Based on the analysis, the point corresponding to \( s \) must lie in the third quadrant to satisfy both conditions \( \sec s < 0 \) and \( \csc s < 0 \).

Key Concepts

Quadrants

In trigonometry, the Cartesian plane is divided into four quadrants. Each quadrant represents a section where the sign of sine, cosine, tangent and their reciprocals, like secant and cosecant, vary. Understanding quadrants is crucial in determining where a particular angle or point lies.

• The first quadrant is where both sine and cosine functions are positive.
• The second quadrant has positive sine and negative cosine functions.
• In the third quadrant, both sine and cosine are negative, making secant and cosecant negative as well.
• The fourth quadrant features negative sine and positive cosine.

When analyzing trigonometric functions, understanding these quadrant characteristics aids in determining the values and signs of the functions.

Secant Function

The secant function, represented as \( \sec s = \frac{1}{\cos s} \), is one of the reciprocal trigonometric functions. It is inherently linked to the cosine function. Knowing the behavior of cosine helps us grasp the behavior of secant.

• When cosine is positive, secant is positive.
• When cosine is negative, secant is also negative.

The analysis of secant becomes predominantly about understanding cosine's behavior in the various quadrants. The problem clearly indicates that secant is negative \((\sec s < 0)\), leading us to initially consider the second and third quadrants, because that's where cosine is negative.

Cosecant Function

The cosecant function, given by \( \csc s = \frac{1}{\sin s} \), is another reciprocal trigonometric function. It is directly associated with the sine function. Grasping sine helps in comprehending cosecant.

• When sine is positive, cosecant is positive.
• When sine is negative, cosecant is negative.

The problem states that cosecant is negative \((\csc s < 0)\), which implies focusing on quadrants where sine is negative — specifically, the third and fourth quadrants.

Cosine

Cosine function, denoted as \(\cos s\), is essential in determining the nature of secant \(\sec s\). The values of cosine vary across quadrants.

• In the first quadrant, \(\cos s\) is positive.
• In the second quadrant, \(\cos s\) is negative.
• In the third quadrant, \(\cos s\) remains negative.
• In the fourth quadrant, \(\cos s\) turns positive again.

Since secant is the reciprocal of cosine, wherever cosine is negative, secant would also be negative, consistent with the exercise's requirement \(\sec s < 0\). This occurs in the second and third quadrants.

Sine

Sine, represented as \(\sin s\), plays a fundamental role in understanding the behavior of the cosecant function \(\csc s\). The value of sine varies across the different quadrants:

• In the first quadrant, \(\sin s\) is positive.
• In the second quadrant, \(\sin s\) remains positive.
• In the third quadrant, \(\sin s\) is negative.
• In the fourth quadrant, \(\sin s\) continues to be negative.

The exercise specifies that \(\csc s\) is negative \((\csc s < 0)\), hence focusing on the third and fourth quadrants where sine is negative provides valuable insight. The third quadrant is where both sine and cosine, and thus cosecant and secant, are negative, fulfilling the condition required.