Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \sin 7 t \sin 3 t $$

Step-by-Step Solution

Answer

\( \sin 7t \sin 3t = \frac{1}{2} ( \cos 4t - \cos 10t ) \).

Step 1: Identify the Trigonometric Identity

We begin by identifying the appropriate trigonometric identity that allows us to express the product of sines as a sum or difference. The relevant identity is:\[\sin A \sin B = \frac{1}{2} \left[ \cos(A - B) - \cos(A + B) \right]\]Here, \(A = 7t\) and \(B = 3t\).

Step 2: Substitute the Values into the Identity

Now, substitute \(A = 7t\) and \(B = 3t\) into the identity:\[\sin 7t \sin 3t = \frac{1}{2} \left[ \cos(7t - 3t) - \cos(7t + 3t) \right]\]

Step 3: Simplify the Expression

Simplify the expression by performing the operations within the cosine functions:\[\cos(7t - 3t) = \cos(4t) \quad \text{and} \quad \cos(7t + 3t) = \cos(10t)\]Substitute back into the identity:\[\sin 7t \sin 3t = \frac{1}{2} \left[ \cos 4t - \cos 10t \right]\]

Step 4: Present the Final Expression

The final expression for the product \(\sin 7t \sin 3t\) as a sum or difference using trigonometric identities is:\[\frac{1}{2} \left( \cos 4t - \cos 10t \right)\]

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are valuable tools in trigonometry that allow us to express the product of two trigonometric functions as a sum or difference of trigonometric functions. This is particularly useful in simplifying expressions, solving equations, and analyzing waveforms in physics and engineering.
For example, the product of two sine functions can be converted to a sum using the identity:

• \[\sin A \sin B = \frac{1}{2} \left[ \cos(A - B) - \cos(A + B) \right]\]

Here, \(A\) and \(B\) are angles, and this transformation from a product to a sum makes it easier to handle complex trigonometric expressions. In the original problem, \(\sin 7t \sin 3t\) was simplified using this formula, leading us to an expression in terms of cosines.

Sine Function

The sine function is a fundamental trigonometric function that arises in the study of angles and oscillations. Most commonly, it is seen in the context of right-angled triangles, where it represents the ratio of the length of the opposite side to the hypotenuse.

• It is denoted as \(\sin(\theta)\), where \(\theta\) is the angle.
• The sine function is periodic with a period of \(2\pi\).

In many areas of science and engineering, the sine function describes wave-like phenomena, such as sound waves and light waves. Knowing how to manipulate and transform functions involving sine is crucial, as seen in using the product-to-sum formulas to handle the product \(\sin 7t \sin 3t\).

Cosine Function

The cosine function is closely related to the sine function and represents another fundamental concept in trigonometry. It is used to describe the projection of an angle onto one of the axes in the unit circle.

• Similar to sine, cosine is defined in terms of the lengths of the sides of a right-angled triangle.
• Specifically, it is the ratio of the adjacent side to the hypotenuse and is denoted as \(\cos(\theta)\).

Like sine, the cosine function is periodic with a period of \(2\pi\).
In the original exercise, the product-to-sum formulas utilize the cosine function to express a compound trigonometric expression in a simpler form. This occurs because the transformation from a product of sines to a difference of cosines involves subtracting and adding angles, underlining the deep connection between these functions.

Angle Addition and Subtraction

Angle addition and subtraction identities are essential tools in trigonometry for simplifying expressions and solving problems involving trigonometric functions. They allow us to break down complex angles into simpler terms, which makes calculations more manageable.
For the cosine function, these identities are:

• Angle Addition: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
• Angle Subtraction: \(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

These identities are particularly useful when working with compound angles, as seen in the exercise where \(7t + 3t\) and \(7t - 3t\) are simplified into more straightforward \(10t\) and \(4t\), respectively.
Ultimately, understanding these identities allows for significant simplification and transformation of trigonometric expressions, which is critical in both academic and practical applications.