Problem 73 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( cos(\frac{\pi}{3} + 2\alpha) \)

Step 1: Identify the Trigonometric Identity

The given expression looks like a version of the double angle identity for cosine, which is \( cos(2A) = cos^2(A) - sin^2(A) \). Here, \( A = \frac{\pi}{6} + \alpha \)

Step 2: Apply the Trigonometric Identity

To simplify, replace the \( cos^2(A) - sin^2(A) \) with \( cos(2A) \). Then, the expression simplifies as follows: \( cos(2(\frac{\pi}{6} + \alpha)) \).

Step 3: Simplify the Expression

Multiplying the argument of the cosine by 2 gives us \( cos(\frac{\pi}{3} + 2\alpha) \).

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry, often abbreviated as 'cos'. It measures the horizontal distance, known in mathematics as the adjacent side, from a given angle in the context of a right-angled triangle. Specifically, for any angle \( \theta \), the cosine is calculated as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. This function is periodic, meaning it repeats its values over regular intervals.

The cosine function has several important properties:

• The range of the cosine function is from -1 to 1.
• It’s an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• The period of the cosine function is \( 2\pi \), indicating that its values repeat every \( 2\pi \) radians.

These properties make cosine incredibly useful in various areas of mathematics, especially when modeling cyclical and wave-like phenomena.

Double Angle Formula

The double angle formulae are crucial techniques in trigonometry that relate trigonometric functions of twice an angle to functions of the angle itself. Specifically, the double angle formulas for cosine express \( \cos(2A) \) in terms of \( \cos(A) \) and \( \sin(A) \):

• \( \cos(2A) = \cos^2(A) - \sin^2(A) \)
• It can also be presented as \( \cos(2A) = 2\cos^2(A) - 1 \)
• Or \( \cos(2A) = 1 - 2\sin^2(A) \)

In this particular problem, the identity \( \cos(2A) = \cos^2(A) - \sin^2(A) \) was directly used to simplify the expression as \( \cos(2(\frac{\pi}{6} + \alpha)) \). Understanding the double angle formulas is critical because they allow complex expressions to be rewritten in simpler forms, which is a great aid in manipulation and solution of equations involving trigonometric terms.

Angle Addition Formulas

Angle addition formulas are used to determine the trigonometric values of a sum or difference of two angles. These formulas are particularly valuable because they allow us to break down calculations into simpler parts. For cosine, the angle addition formula is:
\[\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\]
For the problem at hand, we relied on these formulas implicitly since the expression \( \cos(\frac{\pi}{6}+\alpha) \cos(\frac{\pi}{6}-\alpha) - \sin(\frac{\pi}{6}+\alpha) \sin(\frac{\pi}{6}-\alpha) \) follows the pattern of the cosine addition formula.

Here are some useful points about angle addition formulas:

• They simplify the process of finding the cosine or sine of the sum or difference of two angles.
• They're derived from basic trigonometric identities and provide a method to work with angles that are sums or differences, rather than simple multiples.

By utilizing these formulas, it becomes much easier to handle trigonometric expressions in algebraic manipulation, as well as in practical applications involving waves or oscillations.