Problem 73 - 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}{6}-\alpha\right)

Step-by-Step Solution

Answer

The simplified expression for the given trigonometric expression is \( \frac{1}{2} \)

Step 1: Apply the cosine addition formula

Applying the cosine of sum formula cos(A ± B) = cosAcosB ∓ sinAsinB, \\[ \cos \left(\frac{\pi}{6}+\alpha\right) = \cos \left(\frac{\pi}{6}\right)\cos {\left(\alpha\right)} - \sin \left(\frac{\pi}{6}\right)\sin \left(\alpha \right) \\ \cos \left(\frac{\pi}{6}-\alpha\right) = \cos \left(\frac{\pi}{6}\right)\cos {\left(\alpha\right)} + \sin \left(\frac{\pi}{6}\right)\sin {\left(\alpha \right) } \]

Step 2: Apply the sine addition formula

Applying the sine of sum formula sin(A ± B) = sinAcosB ± cosAsinB, \\[ \sin \left(\frac{\pi}{6}+\alpha\right) = \sin \left(\frac{\pi}{6}\right)\cos {\left(\alpha\right)} + \cos \left(\frac{\pi}{6}\right)\sin \left(\alpha \right) \\ \sin \left(\frac{\pi}{6}-\alpha\right) = \sin \left(\frac{\pi}{6}\right)\cos {\left(\alpha\right)} - \cos \left(\frac{\pi}{6}\right)\sin {\left(\alpha \right) } \]

Step 3: Substitute the values from Step 1 and Step 2

Substitute these values back into the original equation. We get \\[ \cos \left(\frac{\pi}{6}\right)^2 \cos {\left(\alpha\right)}^2 + \sin \left(\frac{\pi}{6}\right)^2\sin \left(\alpha \right)^2 \]

Step 4: Simplify the expression

Simplify the expression, we get \\[ \cos^2 \left(\frac{\pi}{6}\right) - \sin^2 \left(\frac{\pi}{6}\right) \] Because we know that \( \cos(\pi/6) = \sqrt{3}/2 \) and \( \sin(\pi/6) = 1/2 \), we get \\[ \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \]

Key Concepts

Cosine Addition Formula

The cosine addition formula is a helpful tool in trigonometry. It allows us to find the cosine of the sum or difference of two angles. The formula is expressed as:

• For sum: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• For difference: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

In the given problem, the task is to find the expression that results from applying the cosine addition formula to the angles \( \left(\frac{\pi}{6} + \alpha\right) \) and \( \left(\frac{\pi}{6} - \alpha\right) \).
By applying the formula:

• \( \cos \left(\frac{\pi}{6} + \alpha \right) = \cos \left(\frac{\pi}{6}\right) \cos \alpha - \sin \left(\frac{\pi}{6}\right) \sin \alpha \)
• \( \cos \left(\frac{\pi}{6} - \alpha \right) = \cos \left(\frac{\pi}{6}\right) \cos \alpha + \sin \left(\frac{\pi}{6}\right) \sin \alpha \)

These transformations help break down complex trigonometric expressions for easier manipulation, crucial for simplifying trigonometric expressions.

Sine Addition Formula

Similar to cosine, sine also has a formula for the sum and difference of two angles. The sine addition formula enables us to find the sine of such an angle combination. Here's how the formula looks:

• For sum: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• For difference: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

In our exercise, it's essential to apply the formula to simplify sine expressions involving angles \( \left(\frac{\pi}{6} + \alpha \right) \) and \( \left(\frac{\pi}{6} - \alpha \right) \). Using the formula we have:

• \( \sin \left(\frac{\pi}{6} + \alpha\right) = \sin \left(\frac{\pi}{6} \right) \cos \alpha + \cos \left(\frac{\pi}{6} \right) \sin \alpha \)
• \( \sin \left(\frac{\pi}{6} - \alpha \right) = \sin \left(\frac{\pi}{6} \right) \cos \alpha - \cos \left(\frac{\pi}{6} \right) \sin \alpha \)

Applying these expressions is key to representing complex trigonometric equations more manageably.

Simplifying Expressions

Simplifying trigonometric expressions is simpler when using trigonometric identities correctly. After transforming the initial expressions using the cosine and sine addition formulas, you are left with components that can be combined or reduced further.
The exercise challenge is determining values or reducing terms. For example, using known values:

• \( \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
• \( \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \)

With these values, substitution into our expression allows further simplification:

• Calculate \( \cos^2 \left(\frac{\pi}{6}\right) - \sin^2 \left(\frac{\pi}{6}\right) = \left(\frac{3}{4} - \frac{1}{4}\right) = \frac{1}{2} \).

Through these steps, we've taken complex expressions and found a single simplified term. Understanding the simplification process is vital in solving trigonometric problems efficiently.