Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \cos 5 t+\cos 6 t $$

Step-by-Step Solution

Answer

The expression is \( 2 \cos \left( \frac{11t}{2} \right) \cos \left( \frac{t}{2} \right) \).

Step 1: Identify the Trigonometric Identity

We have two cosine terms: \( \cos 5t \) and \( \cos 6t \). To combine these, we use the cosine addition formula for sums, which is \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \).

Step 2: Assign Values to A and B

In the identity, we assign \( A = 5t \) and \( B = 6t \). This will help us in applying the formula correctly.

Step 3: Calculate \( \frac{A+B}{2} \) and \( \frac{A-B}{2} \)

First, compute \( \frac{A+B}{2} = \frac{5t + 6t}{2} = \frac{11t}{2} \). Next, compute \( \frac{A-B}{2} = \frac{5t - 6t}{2} = \frac{-t}{2} \).

Step 4: Substitute into the Identity

Substitute the calculated values into the identity: \( \cos 5t + \cos 6t = 2 \cos \left( \frac{11t}{2} \right) \cos \left( \frac{-t}{2} \right) \).

Step 5: Simplify the Expression

The negative inside the cosine doesn't change the cosine's value (since \( \cos(-x) = \cos(x) \)), so it becomes: \( 2 \cos \left( \frac{11t}{2} \right) \cos \left( \frac{t}{2} \right) \).

Key Concepts

Cosine Addition Formula

The cosine addition formula is a fundamental tool in trigonometry that helps simplify expressions involving the cosine of two angles. When faced with terms like \( \cos A + \cos B \), the formula to use is:

• \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)

This formula essentially transforms the sum of two cosine terms into a product of cosines, which can be easier to work with in certain problems. It comes in handy when simplifying expressions, solving equations, or analyzing wave functions.
Through this transformation, we take advantage of the trigonometric properties like symmetry and periodicity of the cosine function, making it particularly useful when dealing with oscillatory functions. Remember, this formula is often used in various fields such as physics, engineering, and signal processing.

Angle Sum and Difference

Understanding angle sum and difference identities is crucial for mastering trigonometry. These identities allow us to express functions involving sums or differences of angles in terms of functions of individual angles. For example, the sum formula for cosine is:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Similarly, the difference formula is:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These formulas are powerful because they help break down complex expressions into simpler components. They also reveal the interrelationships between trigonometric functions, allowing one to express a trigonometric function of a combined angle in terms of individual angles. To understand these better, keep in mind that they derive from the unit circle and the geometric interpretations of sine and cosine.

Trigonometric Simplification

Trigonometric simplification involves using identities and formulas to rewrite expressions in a more concise and often more useful form. In the original exercise, we began with two separate cosine terms, \( \cos 5t \) and \( \cos 6t \), and used the cosine addition formula to simplify them into:

• \( 2 \cos \left( \frac{11t}{2} \right) \cos \left( \frac{t}{2} \right) \)

The simplification process not only makes an expression more manageable but often reveals its underlying structure. By converting sums into products or applying identities to reduce the number of terms, these simplifications can be particularly useful in calculus, where finding derivatives or integrals of simplified expressions becomes more straightforward.
Simplification also emphasizes properties like periodicity and symmetry in trigonometric functions, often leading to insights that are not immediately apparent in their original forms.