Problem 38 - Calculus Single Variable

Question

Use one or more of the basic trigonometric identities to derive the given identity. \(\sin (\theta) \sin (\phi)=\frac{\cos (\theta-\phi)-\cos (\theta+\phi)}{2}\)

Step-by-Step Solution

Answer

The identity is derived using the product-to-sum identity for sine.

Step 1: Use Product-to-Sum Identity for Sine

The product-to-sum identities relate the product of trigonometric functions to a sum or difference. For sine, the identity is: \[\sin(a)\sin(b)=\frac{1}{2}[\cos(a-b)-\cos(a+b)].\] Set \(a = \theta\) and \(b = \phi\).

Step 2: Apply Values to Identity

Substitute \(a = \theta\) and \(b=\phi\) into the product-to-sum identity:\[\sin(\theta)\sin(\phi) = \frac{1}{2}[\cos(\theta-\phi)-\cos(\theta+\phi)].\]

Step 3: Confirm Derived Equation

The derived expression matches the original identity to be proven: \[\sin(\theta)\sin(\phi) = \frac{\cos(\theta-\phi)-\cos(\theta+\phi)}{2}.\]

Key Concepts

Understanding the Sine Function

The sine function is a fundamental component of trigonometry, closely linked to the geometry of right-angled triangles. In its simplest form, the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. When dealing with angles in standard position on the unit circle, the sine function gives the y-coordinate of the point where the terminal side of the angle intersects the circle.

• Right Triangle Definition: If you have a right triangle, the sine of an angle \( \theta \) is given by \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• Unit Circle Definition: On the unit circle, \( \sin(\theta) \) is the y-coordinate of the point at angle \( \theta \).

The function is periodic with a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians. It is essential in various mathematical and physics applications, where it helps model oscillations and waves.

Exploring Product-to-Sum Identities

Product-to-sum identities are useful trigonometric formulas that simplify the multiplication of trigonometric functions into sums or differences. For sine functions, the specific identity used is:\[\sin(a)\sin(b) = \frac{1}{2}(\cos(a-b) - \cos(a+b))\] This transformation is effective when solving complex trigonometric equations or proving identities.

• Purpose: These identities allow us to express products as manageable sums or differences, making complex calculations more accessible and simplifying integration and other mathematical operations.
• Applications: They are often applied in physics and engineering to solve oscillatory systems and in signal processing.

By substituting specific values, such as \( a = \theta \) and \( b = \phi \), you can verify or derive identities, as shown in the exercise solution.

Decoding the Cosine Function

The cosine function is another cornerstone of trigonometry, offering a way to relate angles and side lengths in triangles. Similar to the sine function, the cosine of an angle in a right triangle is known from its definition as the ratio of the adjacent side to the hypotenuse. In the unit circle scenario, the cosine function provides the x-coordinate of a point on the circle.

• Right Triangle Definition: For a right triangle, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• Unit Circle Definition: On the unit circle, \( \cos(\theta) \) corresponds to the x-coordinate of the point at angles measured from the positive x-axis.

Periodicity and Applications

The cosine function is periodic with a period of \( 2\pi \), making it pivotal in modeling periodic phenomena such as sound and light waves. Its symmetry and relationship with the sine function (\( \cos(\theta) = \sin(\theta + \frac{\pi}{2}) \)) are often used to manipulate expressions and solve trigonometric equations.