Problem 24 - Precalculus

Question

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

Step-by-Step Solution

Answer

The terminal side of \( \theta \) is in the second quadrant.

Step 1: Understanding the Sign of Cotangent

The cotangent of an angle, \( \cot \theta \), is negative when the sine and cosine of the angle have opposite signs. This occurs in the second and fourth quadrants.

Step 2: Understanding the Sign of Cosecant

The cosecant of an angle, \( \csc \theta \), is positive when the sine of the angle is positive. This happens in the first and second quadrants because sine is positive in these quadrants.

Step 3: Finding the Common Quadrant

We need to find a quadrant where \( \cot \theta \) is negative and \( \csc \theta \) is positive. From Steps 1 and 2, the second quadrant is the only quadrant that satisfies both conditions, as cosine is negative and sine is positive there.

Key Concepts

Cotangent Sign

The cotangent function, denoted as \( \cot \theta \), is the reciprocal of tangent. It is calculated as the ratio of the cosine to the sine:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
The sign of \( \cot \theta \) depends on the signs of both \( \cos \theta \) and \( \sin \theta \). Since it is a ratio, \( \cot \theta \) is negative when \( \cos \theta \) and \( \sin \theta \) have opposite signs. This occurs in the following quadrants:

• Second Quadrant: \( \sin \theta > 0 \) and \( \cos \theta < 0 \)
• Fourth Quadrant: \( \sin \theta < 0 \) and \( \cos \theta > 0 \)

These are the quadrants where the terminal side of \( \theta \) can lie to satisfy \( \cot \theta \) being negative. It is crucial to understand the interplay between sine and cosine in these quadrants to determine the cotangent's sign accurately.

Cosecant Sign

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. It is expressed as:
\[ \csc \theta = \frac{1}{\sin \theta} \]
Because it relies on sine, the sign of \( \csc \theta \) is directly influenced by the sign of \( \sin \theta \). Therefore, \( \csc \theta \) is positive when \( \sin \theta \) is positive.

• First Quadrant: \( \sin \theta > 0 \)
• Second Quadrant: \( \sin \theta > 0 \)

The positive sign of \( \csc \theta \) indicates that \( \theta \) must lie in a quadrant where sine is positive, such as the first or the second quadrant.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, relating the angles of a triangle to the lengths of their sides. The primary functions include sine, cosine, tangent, cotangent, secant, and cosecant. Each function has a specific role and behavior in different quadrants, as follows:

• First Quadrant: All trigonometric functions are positive.
  • Sine, cosine, tangent, cotangent, secant, and cosecant are all positive here.
• Second Quadrant: Only sine and cosecant are positive.
  • Cosine, tangent, cotangent, and secant are negative.
• Third Quadrant: Only tangent and cotangent are positive.
  • Sine, cosine, secant, and cosecant are negative.
• Fourth Quadrant: Only cosine and secant are positive.
  • Sine, tangent, cotangent, and cosecant are negative.

Understanding these functions and their signs across quadrants is essential for solving trigonometric problems. It helps determine the correct quadrant location for a given problem, which influences the function's sign and computation.