Problem 94 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determine the sign of tan

The cotangent of an angle, \(\cot \theta\), is the reciprocal of the tangent, \(\tan \theta\). If \( \cot \theta < 0 \), then \( \tan \theta \) must also be negative.

Step 2: Identify quadrants for negative tangent

In the unit circle, \(\tan \theta\) is negative in the second and fourth quadrants, because tangent is positive in the first and third quadrants.

Step 3: Determine the sign of cosine

The secant of an angle, \(\sec \theta\), is the reciprocal of the cosine, \(\cos \theta\). If \( \sec \theta < 0 \), then \( \cos \theta \) must also be negative.

Step 4: Identify quadrants for negative cosine

In the unit circle, \(\cos \theta\) is negative in the second and third quadrants, because cosine is positive in the first and fourth quadrants.

Step 5: Identify common quadrants

Both conditions, \( \tan \theta < 0 \) and \( \cos \theta < 0 \), occur together in the second quadrant. Therefore, \(\theta\) is in the second quadrant.

Key Concepts

Unit Circle

The Unit Circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. This circle provides a way to visualize and understand trigonometric functions and their values at different angles. Each point on the unit circle corresponds to an angle, measured in either degrees or radians, and has coordinates \((\cos(\theta), \sin(\theta))\).

The unit circle helps students see the relationships between the different trigonometric functions.

• \(\sin(\theta)\) corresponds to the y-coordinate, or the vertical distance from the x-axis.
• \(\cos(\theta)\) is the x-coordinate, or the horizontal distance from the y-axis.
• \(\tan(\theta)\) is the slope formed by these coordinates, \(\frac{\sin(\theta)}{\cos(\theta)}\).

Knowing these relationships around the unit circle helps in solving problems, like determining function signs as angles traverse through different quadrants.

Quadrants

The coordinate plane is divided into four sections called quadrants. These quadrants are numbered counterclockwise, starting from the positive x-axis (which is the first quadrant). The orientation is crucial for understanding where each trigonometric function exhibits positive or negative values.

• The First Quadrant: In this section, both sine and cosine of the angle are positive, making all trigonometric functions positive here.
• The Second Quadrant: Sine remains positive, while cosine and tangent become negative.
• The Third Quadrant: Here, tangent is positive, while sine and cosine are negative.
• The Fourth Quadrant: Cosine becomes positive again, but sine and tangent are negative.

Understanding which quadrant an angle lies in helps predict the signs of trigonometric functions and plays a significant role in graphing and analyzing trigonometric equations.

Signs of Trigonometric Functions

The signs of trigonometric functions change depending on the quadrant in which the angle lies. This concept is pivotal in solving equations or conditions based on function values. Here's how it works:

• Sine Function: Sine is positive in the first and second quadrants and negative in the third and fourth.
• Cosine Function: Cosine is positive in the first and fourth quadrants, negative in the second and third.
• Tangent Function: Tangent is positive in the first and third quadrants, negative in the second and fourth.

The exercise requires knowing that if \(\cot \theta < 0\), it implies that \(\tan \theta\) is negative, which occurs in the second and fourth quadrants. Similarly, if \(\sec \theta < 0\), indicating \(\cos \theta\) is negative, such a condition is observed in the second and third quadrants. So, the angle \(\theta\) most likely resides in the second quadrant, where both conditions coexist.