Problem 72 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The simplified equation is \(cos^2(\frac{\pi}{6}) - sin^2(\alpha)\)

Step 1: Identify the Parts of the Expression

Look at the given expression and identify the values \(A = \frac{\pi}{6}\) and \(B = \alpha\) which makes up the angles. The formula is structured as \(cos(A + B) cos (A - B) - sin (A + B) sin (A - B)\)

Step 2: Apply the identity

Apply the cosine of sum and difference of two angles identity to the expression. According to the identity, \(cos(A + B) cos (A - B) - sin (A + B) sin (A - B) = cos^2(A) - sin^2(B)\)

Step 3: Substitute the values of A and B

Substitute the values of A and B into the right side of the equation, which is \(cos^2(A) - sin^2(B)\). Therefore, the expression simplifies to \(cos^2(\frac{\pi}{6}) - sin^2(\alpha)\)

Key Concepts

Cosine of Sum and Difference

The cosine of the sum and difference identity is a fundamental equation in trigonometry. It allows us to express the cosine of a complex angle, such as the sum or difference of two angles, using simpler trigonometric expressions.
In particular, the identity states that:

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

This identity simplifies calculations in trigonometry, especially when dealing with composite angles.
When we encounter expressions that involve angles like the ones in the given exercise, applying this identity helps to transform them into more manageable forms, paving the way for further simplifications. It ensures clarity and leads to precise results, as showcased in the exercise where the expression was reduced to a simpler form using these identities.

Angle Simplification

Angle simplification is the process of making a trigonometric expression as simple as possible, often resulting in a single-term expression.
In trigonometry, many expressions involve angles that can be broken down into their components for easier manipulation and understanding.
In the exercise, the expression involves angles \( \frac{\pi}{6} \) and \( \alpha \). To simplify it, the given angles were identified as part of a structure that could be addressed using known identities.
By applying the appropriate identity from trigonometry, the potentially complicated expression transforms into one that requires less mathematical processing. This approach not only makes it easier to work with the expression but also provides a deeper understanding of the relationship between the angles involved, illustrating the power of angle simplification in mathematical problem-solving.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, describing the relationships between the angles and sides of triangles.
The primary trigonometric functions are sine, cosine, and tangent, each with their respective reciprocal functions such as cosecant, secant, and cotangent.
In the context of the exercise, the focus is on the sine and cosine functions.
The cosine function, specifically, when dealing with the sum and difference of angles, plays a crucial role in simplifying expressions.
Understanding these functions allows for the manipulation of complex expressions and aids in solving mathematical problems involving triangle measurements.
By applying the concepts of these functions, students can simplify expressions, as demonstrated in the example, using a combination of identities and substitutions to deduce an expression involving \(\cos^2\) and \(\sin^2\).
This highlights how foundational understanding of trigonometric functions is essential to conquering more advanced mathematical challenges.