Problem 96 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The angle \( \theta \) lies in the second and fourth quadrants.

Step 1: Understanding Tangent and Cotangent

Recall that the tangent function, \( \tan \theta \), is positive in the first and third quadrants, and negative in the second and fourth quadrants. The cotangent function, \( \cot \theta \), has the same sign as the tangent function since \( \cot \theta = \frac{1}{\tan \theta} \). Thus, if both \( \tan \theta < 0 \) and \( \cot \theta < 0 \), \( \theta \) must lie in a quadrant where both \( \tan \theta \) and \( \cot \theta \) are negative.

Step 2: Identifying Quadrants for Negative Tangent

Based on the information available, where both functions are negative, \( \tan \theta \) (and \( \cot \theta \)) is negative in the second and fourth quadrants. We can now focus on these quadrants as possibilities.

Step 3: Verification

We find that tangent is negative in the second and fourth quadrants. Since \( \cot \theta\) is also negative by condition, it confirms that the angle \( \theta \) lies in these quadrants, since these are the same locations where \( \tan \theta \) and \( \cot \theta \) would match being negative.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of its opposite side to its adjacent side. In every full circle or 360°, the tangent function exhibits a recurring pattern due to the periodic nature of trigonometric functions.
When we look at the unit circle, as the angle \( \theta \) moves through different quadrants, the sign of \( \tan \theta \) changes:

• First Quadrant: \( \tan \theta > 0 \) since both the sine and cosine values are positive.
• Second Quadrant: \( \tan \theta < 0 \) as sine is positive and cosine is negative.
• Third Quadrant: \( \tan \theta > 0 \) with both sine and cosine being negative making the ratio positive.
• Fourth Quadrant: \( \tan \theta < 0 \) since sine is negative and cosine is positive.

Knowing these sign changes across quadrants helps in determining which quadrants a specific condition, such as \( \tan \theta < 0 \), may be satisfied.

Cotangent Function

The cotangent function, \( \cot \theta \), is the trigonometric function defined as the reciprocal of the tangent function, \( \cot \theta = \frac{1}{\tan \theta} \).
This particular function shares many characteristic behaviors with the tangent function, particularly in terms of the sign changes it undergoes within the angle quadrants.
Since \( \cot \theta \) is just the inverse of \( \tan \theta \), it means:

• When \( \tan \theta \) is positive, \( \cot \theta \) is also positive.
• When \( \tan \theta \) is negative, \( \cot \theta \) is negative as well.

Thus, like \( \tan \theta \), \( \cot \theta \) is negative in the second and fourth quadrants. This mirroring of signs is pivotal when analyzing problems where both of these functions need to be negative, as seen in exercises about angle quadrants.

Angle Quadrants

In trigonometry, angle quadrants are a way of dividing the complete circle or 360° into four sections, or quadrants, to easily determine the sign of trigonometric functions such as sine, cosine, tangent, and cotangent. Each quadrant is defined as having a span of 90°, starting from the positive x-axis:

• First Quadrant: Positive angle measures from 0° to 90°, where all trigonometric functions are positive.
• Second Quadrant: Ranges from 90° to 180°, where sine is positive but cosine and tangent are negative.
• Third Quadrant: Spans 180° to 270°, where only tangent is positive, and sine and cosine are negative.
• Fourth Quadrant: Lies between 270° and 360°, where cosine is positive, and sine and tangent are negative.

When we have conditions regarding the negativity of \( \tan \theta \) and \( \cot \theta \), identifying the corresponding quadrants can guide us to the right solution, like knowing a \( \tan \theta < 0 \) suggests quadrants where the functions take negative values.