Problem 98 - A Graphical Approach to Precalculus with Limits

Question

Explain why there is no angle \(\theta\) that satisfies \(\tan \theta>0, \cot \theta<0\)

Step-by-Step Solution

Answer

No angle \(\theta\) satisfies \(\tan \theta>0\) and \(\cot\theta<0\) simultaneously.

Step 1: Understanding the Tangent Condition

The condition \(\tan \theta > 0\) indicates that the angle \(\theta\) must be in either the first quadrant (\(0^\circ < \theta < 90^\circ\)) or the third quadrant (\(180^\circ < \theta < 270^\circ\)). This is because the tangent function is positive in both of these quadrants.

Step 2: Understanding the Cotangent Condition

The condition \(\cot \theta < 0\) means \(\theta\) must be in a quadrant where the cotangent function is negative. Since \(\cot \theta = \frac{1}{\tan \theta}\), \(\cot \theta\) is negative in the second quadrant (\(90^\circ < \theta < 180^\circ\)) and in the fourth quadrant (\(270^\circ < \theta < 360^\circ\)) as it is the reciprocal of a positive tangent.

Step 3: Checking for Common Quadrants

Compare the quadrants where the conditions \(\tan \theta > 0\) and \(\cot \theta < 0\) are satisfied. \(\tan \theta > 0\) is satisfied in the first and third quadrants, while \(\cot \theta < 0\) is satisfied in the second and fourth quadrants.

Step 4: Conclusion

Since there are no common quadrants where both \(\tan \theta > 0\) and \(\cot \theta < 0\), there is no angle \(\theta\) that meets both conditions simultaneously. Therefore, it is impossible for such an angle to exist.

Key Concepts

Tangent Function

The tangent function is a fundamental part of trigonometry and is one of the basic trigonometric functions. It is typically denoted as \( \tan \theta \), where \( \theta \) is an angle in a right triangle. The tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. Mathematically, this is expressed as:\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]This function is periodic with a period of \(180^\circ\) or \(\pi\) radians and is positive in two specific quadrants:

• First Quadrant (\(0^\circ < \theta < 90^\circ\))
• Third Quadrant (\(180^\circ < \theta < 270^\circ\))

In these quadrants, the values of \(\sin \theta\) and \(\cos \theta\) both have the same sign, resulting in a positive tangent. The tangent is especially important in defining angles and explaining the periodic nature of trigonometric functions.

Cotangent Function

The cotangent function is another key concept in trigonometry. It is denoted by \( \cot \theta \) and is the reciprocal of the tangent function. This means that:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}} \]The cotangent function is defined for all angles except those where \( \tan \theta = 0 \), as this would result in division by zero. The period of the cotangent function is also \(180^\circ\) or \(\pi\) radians. Unlike the tangent function, the cotangent is negative in the quadrants where \( \tan \theta \) is positive, which are:

• Second Quadrant (\(90^\circ < \theta < 180^\circ\))
• Fourth Quadrant (\(270^\circ < \theta < 360^\circ\))

In these quadrants, \(\tan \theta\) is negative, so the reciprocal (cotangent) is also negative. Understanding the cotangent involves grasping its reciprocal nature and how it interacts with the tangent function.

Quadrants in Trigonometry

Trigonometry often relies on dividing the coordinate plane into four sections, or quadrants, which helps determine the signs of trigonometric functions based on the angle's location. These quadrants are numbered counter-clockwise starting with the upper right section.

• First Quadrant: Both \(\sin \theta\) and \(\cos \theta\) are positive, so \( \tan \theta > 0 \).
• Second Quadrant: \(\sin \theta\) is positive, \(\cos \theta\) is negative, making \( \tan \theta < 0 \).
• Third Quadrant: Both \(\sin \theta\) and \(\cos \theta\) are negative, so \( \tan \theta > 0 \).
• Fourth Quadrant: \(\sin \theta\) is negative, \(\cos \theta\) is positive, resulting in \( \tan \theta < 0 \).

Understanding which trigonometric functions are positive or negative in each quadrant is essential. This understanding is critical when determining conditions like \(\tan \theta > 0\) or \(\cot \theta < 0\) and finding if there’s a specific angle \(\theta\) that satisfies both.