Problem 25 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \cot ^{2} \theta+\cot \theta=0 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = 90^\circ + 180^\circ n \) and \( \theta = 135^\circ + 180^\circ n \), where \( n \) is any integer.

Step 1: Rewrite the Equation

We start with the equation \( \cot^2 \theta + \cot \theta = 0 \). This can be factored as \( \cot \theta (\cot \theta + 1) = 0 \). This implies that either \( \cot \theta = 0 \) or \( \cot \theta + 1 = 0 \).

Step 2: Solve \(\cot \theta = 0\)

The cotangent is zero wherever the tangent is undefined. Since \( \cot \theta = \frac{1}{\tan \theta} \), \( \tan \theta \) must be undefined. This occurs when \( \theta = 90^\circ + 180^\circ n \), where \( n \) is any integer.

Step 3: Solve \(\cot \theta + 1 = 0\)

This can be rewritten as \( \cot \theta = -1 \). The cotangent is -1 when the angle \( \theta \) corresponds to \( 135^\circ + 180^\circ n \), where \( n \) is any integer. This occurs because cotangent and tangent are periodic with period 180 degrees.

Step 4: Combine Both Solutions

The general solution for the equation includes both sets of solutions found previously. Therefore, the solutions are \( \theta = 90^\circ + 180^\circ n \) and \( \theta = 135^\circ + 180^\circ n \) for any integer \( n \).

Key Concepts

Cotangent Function

The cotangent function is an essential trigonometric function, often abbreviated as \( \cot \). It is the reciprocal of the tangent function, defined as \( \cot \theta = \frac{1}{\tan \theta} \). This relationship shows that cotangent is undefined where tangent is zero, and vice versa. In practical terms, \( \cot \theta \) represents the ratio of the adjacent side to the opposite side in a right triangle. Cotangent is periodic with a period of 180 degrees or \( \pi \) radians, meaning its values repeat every half rotation. When dealing with trigonometric equations involving cotangent, it's vital to recognize the intervals where it is defined and its periodic nature. This understanding can greatly aid in solving equations accurately.

Solving Trigonometric Equations

Solving trigonometric equations like \( \cot^2 \theta + \cot \theta = 0 \) requires a clear strategy. Start by rewriting or factoring the equation, which simplifies the process. For example, this equation factors to \( \cot \theta (\cot \theta + 1) = 0 \). This indicates that we examine each factor separately. So, two scenarios arise: either \( \cot \theta = 0 \) or \( \cot \theta + 1 = 0 \).

• For \( \cot \theta = 0 \), identify when the cotangent of an angle is zero. Since \( \cot \theta = \frac{1}{\tan \theta} \), this happens when tangent is undefined, at angles like \( 90^\circ + 180^\circ n \).
• For \( \cot \theta = -1 \), find when the cotangent of angle equals -1, which occurs at \( 135^\circ + 180^\circ n \). Understanding these solutions involves recognizing cotangent's periodic behavior, with repetition every 180 degrees.

Ultimately, solving involves combining these solutions to understand all potential values of \( \theta \).

Angle Measurement in Degrees

In trigonometry, understanding angle measurement in degrees is critical. Angles in a circle are typically measured either in degrees or radians. One full circle is \( 360^\circ \) or \( 2\pi \) radians. Degrees are a more intuitive measure for many, often used in practical applications like navigation and geometry. When working with trigonometric equations, like the one given, knowing how angles are expressed in degrees helps to pinpoint specific solutions such as where functions like cotangent become zero or take on particular values.

• For instance, an angle of \( 90^\circ \) is a quarter of a circle and represents where tangent is undefined, making cotangent zero.
• Similarly, angles such as \( 135^\circ \) align with specific values for trigonometric functions, often used to solve for repeated patterns or periodicity in problems like \( \cot \theta = -1 \).

This knowledge of angle measurement assists in visualizing the solution and understanding the cyclical nature of trigonometric functions.