Problem 109 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \((2 \sin x+\cos x)^{2}+(2 \cos x-\sin x)^{2}=5\)

Step-by-Step Solution

Answer

The equation is an identity as both sides simplify to 5.

Step 1 - Expand the Squared Terms

First, let's expand each squared term inside the expression:\[(2 \sin x + \cos x)^{2} = (2 \sin x + \cos x)(2 \sin x + \cos x) = 4 \sin^2 x + 4 \sin x \cos x + \cos^2 x\]\[(2 \cos x - \sin x)^{2} = (2 \cos x - \sin x)(2 \cos x - \sin x) = 4 \cos^2 x - 4 \sin x \cos x + \sin^2 x\]

Step 2 - Combine Like Terms

Now, add the two expanded expressions together:\[4 \sin^2 x + 4 \sin x \cos x + \cos^2 x + 4 \cos^2 x - 4 \sin x \cos x + \sin^2 x\]

Step 3 - Simplify the Expression

Combine like terms in the expression from Step 2:1. Combine \( \sin^2 x \) terms: \(4 \sin^2 x + \sin^2 x = 5 \sin^2 x\)2. Combine \( \cos^2 x \) terms: \(4 \cos^2 x + \cos^2 x = 5 \cos^2 x\)3. The \( \sin x \cos x \) terms cancel each other out: \(4 \sin x \cos x - 4 \sin x \cos x = 0\)This simplifies to:\[5 \sin^2 x + 5 \cos^2 x\]

Step 4 - Apply Trigonometric Identity

Recall the fundamental trigonometric identity: \( \sin^2 x + \cos^2 x = 1\).Substitute this identity into the expression:\[5 (\sin^2 x + \cos^2 x) = 5 \times 1 = 5\]

Step 5 - Verify the Identity

Finally, confirm that the identity holds:By simplifying, our left side \((2 \sin x + \cos x)^{2} + (2 \cos x - \sin x)^{2}\) simplifies to 5, which matches the right side of the original equation. Thus, the equation is an identity.

Key Concepts

Algebraic Manipulation

In mathematics, especially in algebra and trigonometry, algebraic manipulation is a critical skill. It involves rearranging and transforming expressions by following algebraic rules. This can include operations such as expanding, factoring, and simplifying expressions. In the exercise provided, the goal of algebraic manipulation is to verify the given trigonometric identity.Let's focus on expanding the squared terms:

• The term \(2 \sin x + \cos x\) is expanded by multiplying it by itself, leading to the expression \(4 \sin^2 x + 4 \sin x \cos x + \cos^2 x\).
• Similarly, \(2 \cos x - \sin x\) is expanded to \(4 \cos^2 x - 4 \sin x \cos x + \sin^2 x\).

This expansion step is crucial as it sets the stage for further simplification. By manipulating these expressions, we transform the complex equation into something more manageable. Always be meticulous with operations to avoid errors, especially when dealing with signs and coefficients. These manipulations allow us to see underlying symmetries or identities that are not immediately obvious.

Simplifying Expressions

The process of simplifying expressions reduces mathematical expressions to their most basic form. This does not change the expression's value but makes it easier to interpret and solve.In our example, after expanding the squared terms, we need to combine like terms:

• The \(\sin^2 x\) terms add up to \(5 \sin^2 x\).
• The \(\cos^2 x\) terms combine to form \(5 \cos^2 x\).
• The \(\sin x \cos x\) terms cancel each other out, equating to zero.

By performing these steps, the expression simplifies significantly, a process that highlights the beauty of algebraic manipulation—making complex expressions simpler and revealing clearer insights into the problem at hand.Always be attentive to details in the simplification process. Canceling opposing terms and combining similar ones efficiently leads to elegant solutions, as seen here.

Fundamental Trigonometric Identity

Trigonometric identities are equations that hold true for all values within their domains. Among these, the fundamental trigonometric identity \(\sin^2 x + \cos^2 x = 1\) is particularly noteworthy.In solving our exercise, after simplification, we reach \(5 \sin^2 x + 5 \cos^2 x\). Leveraging the fundamental identity, we substitute \(\sin^2 x + \cos^2 x\) with 1, transforming the expression into \(5 \times 1 = 5\).This identity not only verifies our original equation but also demonstrates the powerful tool that trigonometric identities provide in simplifying and solving equations.Understanding these identities and knowing when to apply them is crucial:

• They simplify complex-trigonometric expressions.
• They allow verification of trigonometric equations.
• They are pivotal in fields such as engineering, physics, and computer science.

Thus, mastering fundamental identities helps students solve otherwise challenging problems with ease.