Problem 112 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\tan \theta=2, \cot \theta=-2$$

Step-by-Step Solution

Answer

It is impossible for \( \tan \theta = 2 \) and \( \cot \theta = -2 \) to be true simultaneously.

Step 1: Understand the Problem

We need to determine if it's possible for the tangent of an angle to be 2 and its cotangent to be -2 simultaneously. Recall that tangent and cotangent are reciprocal trigonometric functions.

Step 2: Recall Trigonometric Identities

Recall the identity that connects tangent and cotangent: \[ \tan \theta = \frac{1}{\cot \theta} \]Given that \( \tan \theta = 2 \), substitute into the identity to check what \( \cot \theta \) should be.

Step 3: Calculate Cotangent Based on Tangent

If \( \tan \theta = 2 \), then using the identity:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{2} \]The calculated value of \( \cot \theta \) does not equal the given \( \cot \theta = -2 \).

Step 4: Conclude Based on Consistency

Since \( \cot \theta = \frac{1}{2} \) is not equal to \( \cot \theta = -2 \), it is impossible for both \( \tan \theta = 2 \) and \( \cot \theta = -2 \) to be true for the same angle \( \theta \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It is defined as the ratio of the sine of an angle to its cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This function is very important in right-angle trigonometry, especially when dealing with right triangles and unit circles.

In a unit circle, which is a circle with a radius of one, \( \tan \theta \) can be visualized as the length of the line segment from the origin to a point tangent to the circle making an angle \( \theta \) with the positive x-axis.

Another crucial aspect of the tangent function is its periodicity and range. The function repeats every \( 180^\circ \) (or \( \pi \) radians), making it a periodic function. The tangent function can take on any real number as its value, which means its range is all real numbers. This property is useful when solving trigonometric equations involving tangent.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is another fundamental trigonometric function closely related to the tangent. It is defined as the reciprocal of the tangent of an angle:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Understanding \( \cot \theta \) is essential because it provides an alternative perspective on the relationship between the angle's sine and cosine.

Similar to the tangent function, the cotangent function is also periodic. It repeats every \( 180^\circ \) (or \( \pi \) radians), aligning closely with the properties of \( \tan \theta \). However, the crucial difference is that while \( \tan \theta \) is undefined when its denominator, \( \cos \theta \), is zero, \( \cot \theta \) is undefined when \( \sin \theta = 0 \).

In terms of its range, \( \cot \theta \) can also take on any real number value. Therefore, it's useful for problems involving reciprocals of tangents, as it provides additional pathways for solving trigonometric equations.

Reciprocal Functions

Reciprocal trigonometric functions are valuable in extending the concepts of primary trigonometric functions, like sine and cosine, with their inverses or reciprocals.

• For tangent, the reciprocal function is cotangent: \( \cot \theta = \frac{1}{\tan \theta} \)
• Similarly, sine's reciprocal is cosecant: \( \csc \theta = \frac{1}{\sin \theta} \)
• The reciprocal of cosine is secant: \( \sec \theta = \frac{1}{\cos \theta} \)

These reciprocal relationships help when solving more complex trigonometric problems. They are particularly useful when simplifying expressions or checking the validity of trigonometric identities.

Understanding these relations allows us to derive other trigonometric values once we know one primary function's value. This ability often simplifies complex calculations, making it a crucial tool in learning and applying trigonometry effectively. It’s important to realize how these are consistently applied in problems such as verifying identities or solving equations, as they aid in finding solutions that involve converting between various function expressions and verifying consistency.