Problem 92 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The angle \(\theta\) is in the first quadrant.

Step 1: Understanding the Condition for Cosecant

The cosecant function, \(\csc \theta\), is the reciprocal of the sine function. Hence, \(\csc \theta > 0\) holds when \(\sin \theta > 0\). Sine is positive in the first and second quadrants.

Step 2: Understanding the Condition for Cotangent

The cotangent function, \(\cot \theta\), is positive when both sine and cosine have the same sign. This happens in the first quadrant (where both are positive) and in the third quadrant (where both are negative), since \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Step 3: Finding the Intersection of Conditions

From Step 1, \(\csc \theta > 0\) indicates that \ \theta\ can be in the first or second quadrant. From Step 2, \(\cot \theta > 0\) indicates that \ \theta\ can be in the first or third quadrant. Therefore, the only quadrant satisfying both conditions is the first quadrant.

Key Concepts

Quadrants

Trigonometry divides the coordinate plane into four sections known as quadrants. These quadrants are numbered counterclockwise starting from the upper right, or the positive x and y-axis.
In trigonometry, it is crucial to know in which quadrant an angle resides because this determines the sign (+/-) of trigonometric functions like sine, cosine, tangent, and their reciprocals, like cosecant and cotangent.
Each quadrant contains distinct characteristics:

• First Quadrant (I): Both x and y are positive. Therefore, all trigonometric functions have positive values here.
• Second Quadrant (II): x is negative, and y is positive. Only the sine function is positive here, while others including cosine and tangent are negative.
• Third Quadrant (III): Both x and y are negative. In this quadrant, only the tangent and cotangent functions are positive.
• Fourth Quadrant (IV): x is positive, and y is negative. Only cosine and secant are positive.

When identifying the quadrant of an angle that fulfills specific conditions, such as where certain trigonometric functions are positive, it helps to recall these signs of functions in each quadrant.

Cosecant

The cosecant function, denoted as \(\csc \theta\\), is the reciprocal of the sine function.
Understanding this reciprocal relationship is key to solving equations or inequalities involving \(\csc \theta\\).
When \(\csc \theta > 0\\), it implies that \(\sin \theta > 0\\) since \(\csc \theta = \frac{1}{\sin \theta}\\).
This condition is satisfied when the angle \(\theta\\) is in the quadrants where sine is positive. As outlined previously, the sine function is positive in the first and second quadrants.
For students, it is often more intuitive to think about the sine function before considering its reciprocal, cosecant. Once mastered, the reciprocal relationship applies seamlessly to all related problems.

Cotangent

The cotangent function, represented as \( \cot \theta \), is the reciprocal of the tangent function. It can also be expressed as the ratio of cosine to sine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
Cotangent shares a unique property that it is positive in quadrants where both sine and cosine share the same sign.
This condition occurs in two specific quadrants:

• First Quadrant: Here, both sine and cosine are positive, resulting in a positive cotangent.
• Third Quadrant: Both sine and cosine are negative, but their ratio remains positive.

For the given condition \( \cot \theta > 0 \) in the exercise, \( \theta \) could be in either the first or third quadrant.
Remembering this property can help determine the placement of angles when working with cotangent in trigonometric problems.

Sine and Cosine

Sine (\( \sin \theta \)) and cosine (\( \cos \theta \)) are the foundational trigonometric functions based on the unit circle.
The values of these functions are derived from the coordinates of a point on the circle where the terminal side of an angle intersects.

• Sine: Corresponds to the y-coordinate, indicating height at a given angle on the unit circle. It is positive in the first and second quadrants.
• Cosine: Corresponds to the x-coordinate, representing the horizontal distance of a point. This function is positive in the first and fourth quadrants.

Understanding the behavior of these two trigonometric functions is crucial because they determine the behavior of their respective reciprocal functions, cosecant and secant, and affect the positivity or negativity of tangent and cotangent.
By mastering the signs and values of sine and cosine, students can easily solve complex trigonometric equations and determine angle quadrants effectively.