Problem 42 - Precalculus with Calculus Previews

Question

$$ \text { Simplify: } 2 \cos 2 t \cos t-\cos 3 t $$

Step-by-Step Solution

Answer

The simplified expression is \( \cos(t) \).

Step 1: Identity Substitution

We start by utilizing the trigonometric identity for the product of cosines. The identity we will use is:\[ 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \] Substituting \( A = 2t \) and \( B = t \) into the identity gives:\[ 2 \cos 2t \cos t = \cos(3t) + \cos(t) \] Now our expression becomes: \[ \cos(3t) + \cos(t) - \cos(3t) \]

Step 2: Simplify the Expression

Cancel out like terms in the expression:\[ \cos(3t) + \cos(t) - \cos(3t) = \cos(t) \] The \( \cos(3t) \) terms cancel each other, leaving us with \( \cos(t) \).

Key Concepts

Product-to-Sum Formulas

In trigonometry, product-to-sum formulas can be an efficient way to simplify certain expressions involving trigonometric functions. These formulas allow us to express the product of two trig functions as a sum or difference of trig functions. This is particularly useful in cases where products are more difficult to manage than sums or differences.

One of the most commonly used product-to-sum formulas is for the product of two cosine functions:

• For two angles, \( A \) and \( B \), the formula is:\[ 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \]
• This identity is derived from the sum-to-product identities, which themselves stem from the angle addition formulas.

By using this product-to-sum formula, we can convert more complex trigonometric expressions into simpler forms that are easier to manipulate.

Cosine Function

The cosine function, commonly denoted as \( \cos(x) \), is one of the fundamental trigonometric functions. It is defined for a circle centered at the origin with a radius of 1, often called the unit circle. The cosine of an angle \( x \) is the horizontal coordinate of the endpoint of an arc of that angle in the unit circle.

Properties of the cosine function include:

• It is periodic with a period of \( 2\pi \), meaning \( \cos(x + 2\pi) = \cos(x) \).
• The function is even, which implies that \( \cos(-x) = \cos(x) \).
• The range of the cosine function is between -1 and 1, i.e., \(-1 \leq \cos(x) \leq 1 \).

Understanding these properties helps in simplifying and analyzing expressions that include the cosine function. In the given exercise, recognizing how \( \cos(3t) \) interacts with other cosine terms plays a crucial role in simplifying the expression.

Simplifying Expressions

Simplifying trigonometric expressions often involves combining like terms or using identities to rewrite the expression in a more manageable form. In mathematics, the goal is often to create expressions that are both easier to understand and work with. Each step in simplification should maintain equality, ensuring the expression doesn't change its value.

Here are some steps you might commonly use in simplifying expressions:

• **Identify and apply relevant trigonometric identities.** For example, using the product-to-sum identity made it possible to transform the product of two cosines into a sum.
• **Combine and cancel terms.** After transforming the original expression, you must look for terms that can be combined or canceled out. In this exercise, the terms \( \cos(3t) \) appeared on both sides of addition and subtraction, allowing them to cancel each other out.

This methodical approach ensures that you arrive at the simplest form of the trigonometric expression, as seen in reducing the given problem to just \( \cos(t) \).