Problem 60 - Precalculus with Limits

Question

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically. \( \dfrac{\cot \alpha}{\csc \alpha + 1} = \dfrac{\csc \alpha + 1}{\cot \alpha} \)

Step-by-Step Solution

Answer

Once all three steps are completed, it can be confirmed whether the given equation is an identity or not. This confirmation is based on graph matching, table value matching, and algebraic simplification.

Step 1: Graph the Functions

Plot the given functions separately using a graphing utility. This means plotting the left-hand side \( \frac{\cot \alpha}{\csc \alpha + 1} \) as one function and the right-hand side \( \frac{\csc \alpha + 1}{\cot \alpha} \) as another function. If both graphs overlay each other perfectly, this would suggest that the equation is indeed an identity.

Step 2: Use the Table Feature of a Graphing Utility

Use the table feature of a graphing utility to list out a series of values for \( \alpha \) and their corresponding function values for both sides of the equation. Cross-check each pair of function values (LHS vs. RHS). If all the pairs match, this further suggests that the functions are identical.

Step 3: Confirm Algebraically

Now confirm this mathematically. Using basic trigonometric identities we know that \( \csc \alpha = \frac{1}{\sin \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \). Substitute these identities into the original equation which results in \( \frac{\frac{\cos \alpha}{\sin \alpha}}{\frac{1}{\sin \alpha} + 1} = \frac{\frac{1}{\sin \alpha} + 1}{\frac{\cos \alpha}{\sin \alpha}} \). Simplify both sides and if they match then the original equation is an identity.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in understanding the relationship between the angles and sides of triangles. They are heavily used in calculus, physics, and engineering. There are six primary trigonometric functions: sine (\( \sin \alpha \)), cosine (\( \cos \alpha \)), tangent (\( \tan \alpha \)), cosecant (\( \csc \alpha \)), secant (\( \sec \alpha \)), and cotangent (\( \cot \alpha \)).

• **Sine (\( \sin \alpha \))**: Opposite side over hypotenuse in a right-angled triangle.
• **Cosine (\( \cos \alpha \))**: Adjacent side over hypotenuse.
• **Tangent (\( \tan \alpha \))**: Opposite over adjacent side.
• **Cosecant (\( \csc \alpha \))**: Reciprocal of sine, therefore \( \csc \alpha = \frac{1}{\sin \alpha} \).
• **Secant (\( \sec \alpha \))**: Reciprocal of cosine, \( \sec \alpha = \frac{1}{\cos \alpha} \).
• **Cotangent (\( \cot \alpha \))**: Reciprocal of tangent, meaning \( \cot \alpha = \frac{1}{\tan \alpha} = \frac{\cos \alpha}{\sin \alpha} \).

These functions aid in the transformation of complex trigonometric expressions into simpler forms, allowing us to solve equations as shown in the original problem statement.

Graphing Utilities

Graphing utilities are powerful tools that enable us to visualize mathematical equations, thus demonstrating relationships between different functions. In the context of trigonometric identities, they are used to graph both sides of an equation to check for identity by overlap. When using a graphing utility:

• Start by inputting each expression of the equation separately as different functions.
• Generate the graphs for both functions (left-hand side vs. right-hand side).
• Check the graph to see if they overlap completely. If they do, this suggests that the equation could be an identity.

Graphing utilities also have a table feature which can list function values for consecutive inputs. This feature helps in verifying if the expressions equate for several values, reinforcing the potential identity.

Algebraic Confirmation

Algebraic confirmation takes the visual and empirical findings from graphing and supports them with mathematical reasoning. To confirm an equation as an identity, we rely on algebraic manipulations, often involving substitution with fundamental trigonometric identities.Let's break down the algebraic process:

• **Substitute Known Identities**: Use the core trigonometric identities, such as \( \csc \alpha = \frac{1}{\sin \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \).
• **Simplify**: Simplify the equation by performing operations that reduce both sides of the equation to the same expression.
• **Verification**: Ensure that both simplified expressions match entirely. If they do, this confirms that the original expression is indeed a trigonometric identity.

This methodical approach turns a potentially complex verification into a step-by-step simplification task, allowing for clear and definitive confirmation of trigonometric identities.