Problem 81 - Precalculus

Question

Does there exist an angle \(0 \leq \theta<2 \pi\) such that \(\tan \theta=\cot \theta ?\)

Step-by-Step Solution

Answer

Yes, angles are \( \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{3\pi}{4}, \text{and} \frac{7\pi}{4} \).

Step 1: Understanding the Equation

We start with the equation \( \tan \theta = \cot \theta \). Using the definition of cotangent, this can be rewritten as \( \tan \theta = \frac{1}{\tan \theta} \).

Step 2: Simplifying the Equation

Let's multiply both sides of the equation by \( \tan \theta \), yielding \( \tan^2 \theta = 1 \).

Step 3: Solving the Simplified Equation

This equation can be rearranged to \( \tan^2 \theta - 1 = 0 \). Factoring it gives us \( (\tan \theta - 1)(\tan \theta + 1) = 0 \).

Step 4: Finding Solutions to Factored Form

From the factored equation, we get two possible cases: \( \tan \theta - 1 = 0 \) or \( \tan \theta + 1 = 0 \). Solving these gives \( \tan \theta = 1 \) or \( \tan \theta = -1 \).

Step 5: Determining Possible Angles for \( \tan \theta = 1 \)

The angles where \( \tan \theta = 1 \) are \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is an integer. Thus, within the range \( 0 \leq \theta < 2\pi \), the solutions are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).

Step 6: Determining Possible Angles for \( \tan \theta = -1 \)

The angles where \( \tan \theta = -1 \) are \( \theta = \frac{3\pi}{4} + n\pi \). Within the range \( 0 \leq \theta < 2\pi \), the solutions are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).

Key Concepts

Tangents

The tangent function, often abbreviated as \( \tan \), is a fundamental trigonometric function. It is a ratio that compares the length of the opposite side to the adjacent side in a right-angled triangle. This makes it particularly useful in problems involving triangles.

• When using the unit circle, the tangent of an angle \( \theta \) is the length of the line drawn from the origin to a point that intersects the line \( y = x \), extending beyond the unit circle.
• The tangent function is periodic with a period of \( \pi \), meaning \( \tan(\theta + \pi) = \tan(\theta) \).

Understanding the tangent function's properties such as these aids in solving equations involving tangents.

Cotangents

Cotangent, abbreviated as \( \cot \), is another trigonometric function, closely related to tangent. The cotangent of an angle is the reciprocal of the tangent. Therefore, it is defined as:\[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]The cotangent function has special properties:

• It is undefined when \( \tan \theta = 0 \), because division by zero is undefined.
• Like the tangent, \( \cot \theta \) is periodic with a period of \( \pi \).

Grasping how cotangent relates to tangent is key when solving trigonometric equations such as \( \tan \theta = \cot \theta \).

Equation Solving

Solving trigonometric equations requires understanding both the functions involved and algebraic manipulation. In our exercise, the equation \( \tan \theta = \cot \theta \) is transformed using the cotangent definition. Here's a simplified approach:

• Reformulate \( \cot \theta \) as \( \frac{1}{\tan \theta} \), yielding \( \tan \theta = \frac{1}{\tan \theta} \).
• Multiply both sides by \( \tan \theta \), simplifying to \( \tan^2 \theta = 1 \).
• Restructure as \( \tan^2 \theta - 1 = 0 \) and factor to find solutions \( \tan \theta = 1 \text{ or } \tan \theta = -1 \).

This method highlights how understanding function definitions helps simplify and solve equations efficiently.

Angle Solutions

After factoring and solving \( \tan^2 \theta = 1 \), using the solutions \( \tan \theta = \pm1 \) helps find specific angles in the interval \( 0 \leq \theta < 2\pi \). Let's break this down:

• For \( \tan \theta = 1 \), angles take the form \( \theta = \frac{\pi}{4} + n\pi \). Within our interval, the solutions are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).
• For \( \tan \theta = -1 \), angles conform to \( \theta = \frac{3\pi}{4} + n\pi \). The angles that fit in the range are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).

Hence, these four angles are solutions, showcasing how periodicity helps in finding angle solutions.