Problem 32 - Algebra and Trigonometry

Question

\(25-38\) . Find all solutions of the given equation. $$ \cot \theta+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = 135^\circ + n \cdot 180^\circ \) or \( \theta = \frac{3\pi}{4} + n \cdot \pi \).

Step 1: Isolate the Trigonometric Function

First, isolate the trigonometric function by moving constants to the other side:\[\cot \theta = -1\]

Step 2: Identify Angles with Known Cotangent Values

Recall the cotangent function, where \( \cot \theta = \frac{1}{\tan \theta} \). We know that for \( \tan \theta = -1 \),\[ \theta = 135^\circ + n \cdot 180^\circ \text{ or } \theta = \frac{3\pi}{4} + n \cdot \pi \text{ for any integer } n \]

Step 3: Express General Solution

Using the periodicity of the tangent function, the general solutions where \( \cot \theta = -1 \) are:\[\theta = 135^\circ + n \cdot 180^\circ \text{ or } \theta = \frac{3\pi}{4} + n \cdot \pi, \text{ for any integer } n.\]

Step 4: Verify Solutions

Consider that for each solution, substituting in the original equation: \( \cot \theta + 1 = 0 \) simplfies to \( 0 = 0 \). Therefore, each solution satisfies the equation.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is a fundamental trigonometric function that relates to the tangent function. Specifically, \( \cot \theta \) is the reciprocal of the tangent:

• Formula: \( \cot \theta = \frac{1}{\tan \theta} \)

Understanding this function is essential for solving trigonometric equations because it gives us an alternative expression for angles in terms of tangent values.

Since the cotangent is the reciprocal, whenever \( \tan \theta = 0 \), the cotangent is undefined. However, in cases like \( \tan \theta = -1 \), \( \cot \theta \) would equal \(-1\) if the angles are correctly identified. Remember:

• \( \tan \theta \) and \( \cot \theta \) are related through their values.
• \( \cot \theta = -1 \) indicates that \( \tan \theta \) must also be \(-1\).

This relationship helps identify special angles where these values occur, such as \(135^\circ\) or \(\frac{3\pi}{4}\) radians.

General Solution

In trigonometry, finding a general solution means identifying all possible angles that satisfy a given trigonometric equation.

After isolating the trigonometric function, \( \cot \theta = -1 \), you find angles where this equation holds true. You can determine these angles by considering the known values of \( \tan \theta \), knowing \( \cot \theta = \frac{1}{\tan \theta} \) implies \( \tan \theta = -1 \).

• Exact angles: \(135^\circ\) or \(\frac{3\pi}{4}\)

The goal is not just to find one angle but all angles that meet this condition. Therefore, using the periodic property of trigonometric functions, you consider:

• \(\theta = 135^\circ + n \cdot 180^\circ\) or \(\theta = \frac{3\pi}{4} + n \cdot \pi\)

where \(n\) is any integer. This formula accounts for the infinite number of angles that translate to the given solution by adding the periodic cycle of \(180^\circ\) or \(\pi\) radians.

Periodicity of Tangent

One of the main properties of the tangent (and cotangent) is periodicity. This means after a certain angle, the function values repeat themselves.

The periodicity of the tangent function is crucial in understanding how trigonometric functions behave over different intervals. Specifically:

• The period of \( \tan \theta \) and consequently \( \cot \theta \) is \(180^\circ\) or \(\pi\) radians.

This implies that every \(180^\circ\) or \(\pi\) radians, the value of \( \tan \theta \) repeats. Thus, whenever you find an angle that satisfies \( \cot \theta = -1 \), adding \(n \cdot 180^\circ\) or \(n \cdot \pi\) will lead to another valid angle.

Understanding periodicity allows you to quickly identify a whole set of solutions rather than just relying on a single one, encapsulating all possible solutions within one general expression.