Problem 4 - Contemporary Precalculus

Question

In Exercises \(1-6,\) determine the quadrant containing the termi. nal side of an angle of t radians in standard position under the given conditions. $$\begin{aligned} &\csc t<0 \quad \text { and }\\\ &\sec t>0 \end{aligned}$$

Step-by-Step Solution

Answer

Answer: 4th quadrant

Step 1: Analyzing the conditions

We are given two conditions: \(\csc t < 0\) and \(\sec t > 0\). Let's analyze each condition separately to find the possible quadrants where the angle might lie.

Step 1a: Analyzing \(\csc t < 0\)

Recall that \(\csc t = \frac{1}{\sin t}\). From the condition \(\csc t < 0\), we can say that the sine of the angle has an opposite sign. Therefore, \(\sin t\) must be negative. Considering the quadrants in the standard position, the sine function is negative in the 3rd and the 4th quadrants.

Step 1b: Analyzing \(\sec t > 0\)

Recall that \(\sec t = \frac{1}{\cos t}\). From the condition \(\sec t > 0\), we can say that the cosine of the angle has the same sign. Therefore, \(\cos t\) must be positive. Considering the quadrants in the standard position, the cosine function is positive in the 1st and the 4th quadrants.

Step 2: Combining the results

We have found that the 3rd and 4th quadrants result in a negative sine, and the 1st and 4th quadrants result in a positive cosine. The only quadrant that satisfies both conditions is the 4th quadrant. So, the quadrant containing the terminal side of an angle of \(t\) radians in standard position under the given conditions is the 4th quadrant.

Key Concepts

Understanding Quadrants in Trigonometry

In trigonometry, there are four quadrants in the coordinate plane, each representing different ranges for angles. Each quadrant is defined by the sign of the trigonometric functions (sine and cosine) in that region.

• The 1st quadrant includes angles from 0 to 90 degrees (0 to \(\frac{\pi}{2}\) radians), where both sine and cosine are positive.
• The 2nd quadrant covers angles from 90 to 180 degrees (\(\frac{\pi}{2}\) to \(\pi\) radians), where sine is positive and cosine is negative.
• The 3rd quadrant spans from 180 to 270 degrees (\(\pi\) to \(\frac{3\pi}{2}\) radians), where both sine and cosine are negative.
• The 4th quadrant includes angles from 270 to 360 degrees (\(\frac{3\pi}{2}\) to \(2\pi\) radians), where sine is negative and cosine is positive.

Understanding these quadrants is crucial in determining the sign of trigonometric functions and solving various problems.

The Cosecant Function Explained

The cosecant function, represented as \(\csc\), is the reciprocal of the sine function. It is defined as \(\csc t = \frac{1}{\sin t}\). Hence, whenever sine is positive, cosecant remains positive, and vice versa.

In practical terms, this means:

• \(\csc t\) is negative when \(\sin t\) is negative, specifically in the 3rd and 4th quadrants.
• Similarly, \(\csc t\) is positive in the 1st and 2nd quadrants where \(\sin t\) is positive.

Working with \(\csc\) involves understanding its relationship with sine, allowing us to assess angle positions and functions within different quadrants.

Secant Function and its Properties

The secant function, denoted by \(\sec\), is the reciprocal of the cosine function, defined as \(\sec t = \frac{1}{\cos t}\). It reflects properties similar to those of cosine but in a reciprocal way.

Here's how it works:

• \(\sec t\) is positive when \(\cos t\) is positive, which occurs in the 1st and 4th quadrants.
• Conversely, \(\sec t\) is negative when \(\cos t\) is negative, which is true in the 2nd and 3rd quadrants.

By understanding secant's behavior with cosine, identifying the correct quadrant where an angle resides becomes more intuitive.

Determining Angle Positions

Determining the position of angles involves understanding the intersection of conditions described by trigonometric functions within the quadrants. Given the conditions \(\csc t < 0\) and \(\sec t > 0\):

• \(\csc t < 0\) implies \(\sin t\) is negative, placing the angle in either the 3rd or 4th quadrant.
• \(\sec t > 0\) indicates \(\cos t\) is positive, suggesting the angle lies in the 1st or 4th quadrant.

The overlap of these conditions occurs in the 4th quadrant, where both criteria are satisfied. Here, \(\sin t\) is negative and \(\cos t\) is positive, making it the sole quadrant fulfilling both conditions. This understanding helps in accurately identifying angle positions.