Problem 83 - Algebra and Trigonometry

Question

(a) Graph \(f(x)=\frac{\sin 3 x}{\sin x}-\frac{\cos 3 x}{\cos x}\) and make a conjecture. (b) Prove the conjecture you made in part (a).

Step-by-Step Solution

Answer

The function \(f(x) = 2\) everywhere it's defined.

Step 1: Analyze the expression

Let's analyze the function given: \(f(x) = \frac{\sin 3x}{\sin x} - \frac{\cos 3x}{\cos x}\). We should simplify or rewrite the expression to help with the graphing process.

Step 2: Simplify each term

Using the trigonometric identities for triple angles: \(\sin 3x = 3\sin x - 4\sin^3 x\) and \(\cos 3x = 4\cos^3 x - 3\cos x\), we can substitute these into the original expression to simplify.

Step 3: Simplify \(\frac{\sin 3x}{\sin x}\)

Substitute \(\sin 3x\) with the identity: \(\frac{\sin 3x}{\sin x} = 3 - 4\sin^2 x\). Similarly, simplify \(\frac{\cos 3x}{\cos x}\).

Step 4: Simplify \(\frac{\cos 3x}{\cos x}\)

Substitute \(\cos 3x\) with the identity: \(\frac{\cos 3x}{\cos x} = 4\cos^2 x - 3\). Thus, the expression becomes \(f(x) = (3 - 4\sin^2 x) - (4\cos^2 x - 3)\).

Step 5: Combine terms

Combine the two simplified terms: \(f(x) = 3 - 4\sin^2 x - 4\cos^2 x + 3\). Recall the identity \(\sin^2 x + \cos^2 x = 1\).

Step 6: Simplify using Pythagorean Identity

Substitute the identity into the expression: \(f(x) = 6 - 4\), which simplifies to \(f(x) = 2\). Therefore, \(f(x)\) is actually a constant function.

Step 7: Conjecture based on the graph

Upon graphing \(f(x)\), observe that the function is a horizontal line at \(y = 2\). Therefore, the conjecture is that \(f(x) = 2\) for all \(x\), except where the original function has discontinuities (undefined points such as multiples of \(\frac{\pi}{2}\)).

Step 8: Prove the conjecture

Since we have algebraically simplified \(f(x)\) to the constant 2, this proves that except for singularities due to division by zero in the original expression, \(f(x)\) is indeed \(2\) for all \(x\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the equation are defined. They are essential tools in simplifying expressions and solving equations in trigonometry. In the given problem, the triple angle identities are used to simplify the function:

• Sine Triple Angle Identity: \[ \sin 3x = 3\sin x - 4\sin^3 x \]This identity helps to express \( \sin 3x \) in terms of \( \sin x \), which can simplify the function \( \frac{\sin 3x}{\sin x} \).
• Cosine Triple Angle Identity: \[ \cos 3x = 4\cos^3 x - 3\cos x \]Similarly, this allows for \( \cos 3x \) to be rewritten in terms of \( \cos x \), simplifying \( \frac{\cos 3x}{\cos x} \).

Understanding and using these identities correctly can greatly reduce the complexity of expressions involving trigonometric functions. This ability to transform expressions is key in both proving conjectures and simplifying computational steps, guiding us to clearer insights such as identifying when a complex-looking expression is actually a constant value.

Graphing Functions

When graphing trigonometric functions, understanding their periodic behavior and transformations is crucial. For the function given in the problem, the goal was to graph:\[ f(x) = \frac{\sin 3x}{\sin x} - \frac{\cos 3x}{\cos x} \]By simplifying this to a constant, \( f(x) = 2 \), we learn about the behavior of the function.

• Step-by-step transformation: Starting with a complex expression and reducing it to a simple constant is a great example of utilizing algebraic manipulation for easier graphing.
• Graphical Insights: A constant graph, like \( y = 2 \), appears as a horizontal line which confirms that outside of any undefined points, the value does not change regardless of \( x \).

Plotting such functions manually or using graphing software supports the verification process, letting us visually confirm that the function consistently upholds a value and validates conjectures based on algebraic simplification.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental and widely used trigonometric identities. It states that:\[ \sin^2 x + \cos^2 x = 1 \]This identity plays a pivotal role in many trigonometric simplifications, including the given problem.

• Simplification Utility: By recognizing and substituting this identity, the expression \( 4\sin^2 x + 4\cos^2 x \) simplifies directly to \( 4 \times 1 = 4 \) in our example.
• Proof Development: Using such fundamental identities enhances the rigor of proving conjectures, as in transforming our graph analysis into mathematical certainty by showing that the expression becomes simply 6 - 4.

Understanding and applying the Pythagorean identity allows for the simplification of trigonometric expressions which can initially seem very complex, leading to simpler forms like the constant value resulting from these transformations. It also provides a reliable basis for ensuring calculations are correct throughout various transformations and manipulations in trigonometric problems.