Problem 31 - 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. $$ \cot (x / 2)=1 $$

Step-by-Step Solution

Answer

The solutions for \(x\) are \(x = \frac{\pi}{2} + 2n\pi\), where \(n\) is an integer.

Step 1: Convert Cotangent to Tangent

Recall the cotangent function is the reciprocal of the tangent function. Therefore, \[\cot \left( \frac{x}{2} \right) = 1 \, \Rightarrow \, \frac{1}{\tan \left( \frac{x}{2} \right)} = 1.\]This means \(\tan \left( \frac{x}{2} \right) = 1.\)

Step 2: Solve for Angle in Terms of Pi

We know that when \(\tan\) is equal to 1, the angle can be either \(45^{\circ}\) or \(225^{\circ}, 405^{\circ}, \ldots\), because the tangent repeats every \(180^{\circ}\). This translates in radian measure as\[\frac{x}{2} = \frac{\pi}{4} + n\pi,\]where \(n\) is an integer.

Step 3: Solve for x

Multiply every term in the equation from Step 2 by 2 to solve for \(x\):\[x = \frac{\pi}{2} + 2n\pi.\]This equation gives us the general solution for \(x\). This means \(x\) is \( \frac{\pi}{2} \) plus any multiple of \(2\pi\).

Key Concepts

Cotangent Function

Understanding the cotangent function is crucial in solving trigonometric equations involving it. The cotangent function, denoted as \( \cot \theta \), is the reciprocal of the tangent function. It means that:

• \( \cot \theta = \frac{1}{ \tan \theta} \)
• It is usually undefined for angles where the tangent equals zero because dividing by zero is impossible.

Cotangent values are most common in right triangles and periodic functions. This function repeats every \(180^{\circ} \) or \(\pi\) radians, just like the tangent function. So, when you see an equation like \( \cot(x/2) = 1 \), it indicates that the angle, when doubled, positions the tangent function at \(1\) or its equivalent angles, allowing us to leverage our knowledge about tangents to find solutions.

Tangent Function

The tangent function is vital for understanding a wide range of trigonometric equations. It is typically denoted by \( \tan \theta \) and defined as the ratio of the sine function over the cosine function:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• The function has a period of \(180^{\circ}\) or \(\pi\) radians, meaning it repeats every half-circle.

Whenever \( \tan \theta = 1 \), this signals that both sine and cosine are equal, which occurs at angles like \(45^{\circ}\) and \(225^{\circ}\). In the context of trigonometric equations, once identified, these angles can help find solutions by generating a series of angles through its periodic nature. For example, translating these into radian measure creates sequences that help in forming the general equation.

General Solution of Trigonometric Equations

The general solution of trigonometric equations refers to finding all possible angles that satisfy a given equation. The key to solving these equations is recognizing patterns and using periodic properties of trigonometric functions. For the tangent function, because it has a period of \(180^{\circ}\) or \(\pi\) radians, this impacts how we find the general solution.

• For example, if \( \tan \left( \frac{x}{2} \right) = 1 \), it confirms that \( \frac{x}{2} \) aligns with angles like \( \frac{\pi}{4} \), and adding multiples of \(\pi\) will yield other valid solutions.
• This approach helps us write expressions like \( \frac{x}{2} = \frac{\pi}{4} + n\pi \), where \(n\) is any integer.

To isolate \(x\), each term is typically adjusted, as seen where \(x\) becomes \( \frac{\pi}{2} + 2n\pi \). Such equations are incredibly powerful as they represent infinite solutions spaced regularly, crucial for covering all possible angles that meet the initial equation.