Problem 15 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(\theta\) represents an angle measured in degrees. $$ 1+\cot \theta=0 $$

Step-by-Step Solution

Answer

The solutions are \(\theta = 135^\circ + n \times 180^\circ\) and \(\theta = 315^\circ + n \times 180^\circ\), where \(n\) is an integer.

Step 1: Understanding the Equation

The original equation given is \(1 + \cot \theta = 0\). We want to solve for \(\theta\) where \(\theta\) is in degrees.

Step 2: Isolating \(\cot \theta\)

Subtract 1 from both sides of the equation to obtain \(\cot \theta = -1\).

Step 3: Solving for \(\theta\) in Special Angles

Remember that the cotangent function, \(\cot \theta\), is the reciprocal of the tangent function. Therefore, \(\cot \theta = -1\) is equivalent to \(\tan \theta = -1\).

Step 4: Finding Specific Values

The angle \(\theta\) where \(\tan \theta = -1\) is a known special angle. \(\tan 135^\circ = -1\) and \(\tan 315^\circ = -1\).

Step 5: General Solution for \(\theta\)

The general solution for \(\theta\) considering the periodicity of the tangent function \(\tan \theta\) (which has a period of \(180^\circ\)) can be expressed as \(\theta = 135^\circ + n\times 180^\circ\) or \(\theta = 315^\circ + n\times 180^\circ \), where \(n\) is any integer.

Key Concepts

cotangent

Cotangent, often written as \( \cot \theta \), is one of the six fundamental trigonometric functions. In a right triangle, cotangent relates the length of the adjacent side to the length of the opposite side. It can also be defined in terms of sine and cosine:

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

Therefore, understanding the relationship between cotangent and tangent is crucial, as it allows you to reframe and solve equations involving \( \cot \theta \) by leveraging the properties of the tangent function.

tangent function

The tangent function, denoted as \( \tan \theta \), is another essential trigonometric function. It represents the ratio of the opposite side to the adjacent side in a right triangle and can be articulated as:

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

The tangent function is periodic, meaning it repeats its values in regular intervals. Specifically, for \( \tan \theta \), this period is \( 180^\circ \). This property is especially useful when solving trigonometric equations, as it helps establish general solutions by accounting for all possible solutions over regular intervals.

special angles

Special angles in trigonometry are commonly known angles where the trigonometric functions take simple, well-known values. These include angles like \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), \( 90^\circ \), and their equivalents in radians. Knowing the trigonometric values of these angles is very helpful when solving equations. For example:

• \( \tan 45^\circ = 1 \)
• \( \tan 135^\circ = -1 \)
• \( \tan 315^\circ = -1 \)

These angles can be used to quickly identify solutions to equations involving tangent or cotangent values.

general solution

The concept of a general solution in trigonometric equations involves finding all possible angles \( \theta \) that satisfy the given equation. Since trigonometric functions are periodic, their values repeat at regular intervals, allowing for multiple solutions. For the tangent function, these solutions occur every \( 180^\circ \), as it completes one whole cycle at this interval. Thus, for an equation like \( \tan \theta = -1 \), the general solution can be expressed as:

• \( \theta = 135^\circ + n \times 180^\circ \)
• \( \theta = 315^\circ + n \times 180^\circ \)

Here, \( n \) is any integer, acknowledging the periodic nature of tangent and ensuring that all angle solutions are included.