Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ \sin x+\cos x \cot x=\csc x $$

Step-by-Step Solution

Answer

Solutions are all \(x\) in \([0, 2\pi)\) excluding \(x = 0\) and \(x = \pi\).

Step 1: Understand the Equation

The equation is \( \sin x + \cos x \cot x = \csc x \). We'll rework this equation to find solutions within the interval \([0, 2\pi)\). Notice that \(\cot x = \frac{\cos x}{\sin x}\) and \(\csc x = \frac{1}{\sin x}\). We'll use these identities to simplify the equation.

Step 2: Substitute Trigonometric Identities

Substitute \(\cot x\) and \(\csc x\) with their identities: \(\sin x + \cos x \cdot \frac{\cos x}{\sin x} = \frac{1}{\sin x}\). This simplifies to \(\sin x + \frac{\cos^2 x}{\sin x} = \frac{1}{\sin x}\). Multiply through by \(\sin x\) to eliminate fractions.

Step 3: Simplify the Equation

Multiply each term by \(\sin x\): \(\sin^2 x + \cos^2 x = 1\). Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to confirm that this simplifies to \(1 = 1\), which is always true. This means every \(x\) in the interval is a solution, except where the trigonometric functions are undefined.

Step 4: Account for Undefined Points

Identify where the original equation is undefined. This occurs when \(\sin x = 0\), which happens at \(x = 0\) and \(x = \pi\) within the interval \([0, 2\pi)\). Thus, these points are not solutions because the expression would contain divisions by zero.

Key Concepts

Trigonometric Identities

Trigonometric identities are key to solving many equations in trigonometry. In this exercise, we start with the equation \(\sin x + \cos x \cot x = \csc x\). Understanding the trigonometric identities will help us simplify expressions and solve the equation.
Key identities used here include:

• \(\cot x = \frac{\cos x}{\sin x}\)
• \(\csc x = \frac{1}{\sin x}\)
• The Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\)

These identities allow us to rewrite and simplify the equation step by step. By substituting \(\cot x\) and \(\csc x\) with \(\frac{\cos x}{\sin x}\) and \(\frac{1}{\sin x}\), respectively, the equation becomes easier to handle.
This simplification process is crucial as it helps reveal the structure of the equation and leads to easier manipulation. With identities, complex equations become manageable, making trigonometric problem-solving more straightforward.

Interval Solutions

When solving trigonometric equations, finding solutions within a specific interval is often required. For this task, we seek solutions in the interval \([0, 2\pi)\). This interval covers one full cycle of the sine and cosine functions.
After simplifying the given equation \(\sin x + \frac{\cos^2 x}{\sin x} = \frac{1}{\sin x}\), by multiplying through by \(\sin x\), we get \(\sin^2 x + \cos^2 x = 1\).
This equation ultimately resolves to \(1 = 1\), a tautology, meaning every \(x\) in the determined interval is technically a solution. However, the trick is to stay attentive to where the original equation might be undefined, ensuring all viable solutions align with reality.

• Ensure that the solutions are within the specified interval.
• Always take note of any restrictions placed on the variable by the equation.

Undefined Points

When working with trigonometric functions, always be aware of undefined points. In the equation \(\sin x + \cos x \cot x = \csc x\), undefined points occur when there is a division by zero.
In our case, this happens when \(\sin x = 0\) because both \(\cot x\) and \(\csc x\) involve \(\sin x\) in the denominator. Within the interval \([0, 2\pi)\), \(\sin x\) is zero at \(x = 0\) and \(x = \pi\).
These are critical points to consider:

• \(x=0\)
• \(x=\pi\)

At these points, the expressions for \(\cot x\) and \(\csc x\) would be undefined, making \(x=0\) and \(x=\pi\) not valid solutions. Knowing where divisions by zero happen ensures we only accept solutions that make sense mathematically.