Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \cos 4 t \sin 6 t $$

Step-by-Step Solution

Answer

\( \cos 4t \sin 6t = \frac{1}{2} (\sin 10t + \sin 2t) \).

Step 1: Identify the Product-to-Sum Formula

The expression \( \cos 4t \sin 6t \) can be converted into a sum or difference using the product-to-sum identity. The specific product-to-sum identity for \( \cos A \sin B \) is:\[\cos A \sin B = \frac{1}{2} \left( \sin(A+B) - \sin(A-B) \right)\]We will use this identity in the next step.

Step 2: Assign Values to A and B

In the given expression \( \cos 4t \sin 6t \), we assign \( A = 4t \) and \( B = 6t \). These values will be used to apply the product-to-sum formula.

Step 3: Apply the Product-to-Sum Identity

Substitute \( A = 4t \) and \( B = 6t \) into the product-to-sum identity:\[\cos 4t \sin 6t = \frac{1}{2} \left( \sin((4t) + (6t)) - \sin((4t) - (6t)) \right)\]Simplify the expression inside the sine functions.

Step 4: Simplify the Expression

Calculate the expressions \( 4t + 6t \) and \( 4t - 6t \):\[4t + 6t = 10t\]\[4t - 6t = -2t\]Now, substitute these into the equation:\[\cos 4t \sin 6t = \frac{1}{2} \left( \sin 10t - \sin (-2t) \right)\]Recall that \( \sin(-x) = -\sin(x) \), so simplify further.

Step 5: Simplify Using Sine Identity

Apply the identity \( \sin(-x) = -\sin(x) \):\[\cos 4t \sin 6t = \frac{1}{2} \left( \sin 10t + \sin 2t \right)\]This simplifies the given expression as a sum.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another. These identities help simplify complex trigonometric expressions and are crucial in solving trigonometry problems. Two important sets of identities are the Pythagorean identities and the Product-to-Sum identities.

The Product-to-Sum identities transform products of trigonometric functions into sums or differences, making them easier to work with. For example, the identity \( \cos A \sin B = \frac{1}{2} ( \sin(A+B) - \sin(A-B) ) \) was used in our problem to convert the expression \( \cos 4t \sin 6t \) into a more manageable form. Understanding these identities enables students to simplify expressions and solve equations efficiently.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is the art of turning a complicated expression into a simpler one, often by using trigonometric identities. In our example, we started with the product \( \cos 4t \sin 6t \). By recognizing the applicable Product-to-Sum identity, we transformed it into a simple sum: \( \frac{1}{2} ( \sin 10t + \sin 2t ) \).

Steps to simplify include:

• Identify applicable identities that match parts of the expression.
• Assign specific values from the expression to variables in the identity.
• Substitute and simplify using known trigonometric rules, such as \( \sin(-x) = -\sin(x) \).

This process is particularly helpful in calculus and physics, where simplified expressions can make integrals and derivatives easier to calculate.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate angles in right-angle triangles to the ratios of their sides. In the context of our exercise, the functions \( \cos \) and \( \sin \) were central to expressing a product as a sum.

Understanding these functions allows us to manipulate and convert different forms of expressions depending on the application. For example, \( \sin \) and \( \cos \) are periodic functions with specific properties:

• They have a domain of all real numbers and a range of \([-1,1]\).
• They repeat their values in a predictable cycle known as the period.
• The sine function is odd, which means \( \sin(-x) = -\sin(x) \), a crucial property used in our simplification process.

Mastery of these functions and their properties is foundational in both pure and applied mathematics.