Problem 74 - Precalculus Essentials

Question

Rewrite each expression as a simplified expression containing one term.s \(\cos \left(\frac{\pi}{6}+\alpha\right) \cos \left(\frac{\pi}{6}-\alpha\right)-\sin \left(\frac{\pi}{6}+\alpha\right) \)\sin \left(\frac{\pi}{3}-\alpha\right) \cos \left(\frac{\pi}{3}+\alpha\right)+\cos \left(\frac{\pi}{3}-\alpha\right) \sin \left(\frac{\pi}{3}+\alpha\right)$ (Do not use four different identities to solve this exercise.)

Step-by-Step Solution

Answer

\(\frac{1}{2}\cos(\frac{\pi}{3}) + \frac{1}{2}\cos(2\alpha) + \sin(\frac{2\pi}{3})\)

Step 1: Utilize Trigonometric Sum-to-product Formulas

Recognize that the first two expressions are of the form \(\cos(A + B)\cos(A - B)\). The product-to-sum identity can be applied to these terms. Apply the formula: \(\cos(\frac{\pi}{6} + \alpha)\cos(\frac{\pi}{6} - \alpha) = \frac{1}{2}(\cos(2 \cdot \frac{\pi}{6}) + \cos(2\alpha))\). This would result in \(\frac{1}{2}(\cos(\frac{\pi}{3}) + \cos(2\alpha))\). Notice that the \(\cos(\frac{\pi}{3})\) term does not simplify further.

Step 2: Identify Identity

Identify the other two terms which utilizes the identity \(\sin(A)\cos(B) + \cos(A)\sin(B) = \sin(A + B)\). In this case, A would be \((\frac{\pi}{3} - \alpha)\) and B would be \((\frac{\pi}{3} + \alpha)\). Therefore, the terms simplify to \(\sin(2 \cdot \frac{\pi}{3})\). The final simplified term would hence be \(\sin(\frac{2\pi}{3})\).

Step 3: Final Simplified Expression

Combine the results from step 1 and step 2. The final simplified expression would then be \(\frac{1}{2}\cos(\frac{\pi}{3}) + \frac{1}{2}\cos(2\alpha) + \sin(\frac{2\pi}{3})\). All the original expressions have now been simplified into one term

Key Concepts

Sum-to-Product Identities

Sum-to-product identities help in transforming trigonometric expressions involving sums or differences of angles into a product of functions. These identities are useful when simplifying complex expressions, like the one in our exercise.
One common identity is:

• \(\cos(A + B)\cos(A - B) = \frac{1}{2}[\cos(2A) + \cos(2B)]\)

By using this identity, we can break down and simplify the given trigonometric functions. This conversion often makes it easier to combine terms, leading to a simpler and more concise expression.

Simplifying Expressions

Simplifying trigonometric expressions involves reducing them to a form with fewer terms or functions. It often involves applying identities like sum-to-product or product-to-sum.
Here's how it works:

• Look for recognizable patterns or forms, such as sum or difference of angles.
• Apply relevant identities to convert the expression.

In our example, using the sum-to-product identity transforms the given expression into simpler terms. Simplifying isn't just about reducing the number of terms; it's also about making complex-looking functions more manageable and easier to compute.

Trigonometric Simplification

Trigonometric simplification is crucial for solving complex problems in math and physics. It turns expressions with multiple trigonometric terms into something more manageable.
Here’s how you do it effectively:

• Identify which trigonometric identities apply to the specific terms you are simplifying.
• Rewrite terms using these identities, reducing the expression to its most simple form.

In the given problem, once you apply the appropriate identities, you combine the results to present a single term. This process underscores the elegance of trigonometric simplification: taking expressions that initially seem overwhelming and breaking them down into understandable pieces.