Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as a product of trigonometric functions or values. $$\sin 102^{\circ}-\sin 95^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 102^{\circ} - \sin 95^{\circ} = 2 \cos(98.5^{\circ})\sin(3.5^{\circ})\)."}

Step 1: Identify the Formula

To express \(\sin 102^{\circ} - \sin 95^{\circ}\) as a product, we can use the trigonometric identity for the difference of sines: \(\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)\).

Step 2: Calculate the Average of Angles

First, find the average of the angles \(A = 102^{\circ}\) and \(B = 95^{\circ}\). \[\frac{102^{\circ} + 95^{\circ}}{2} = 98.5^{\circ}\]

Step 3: Calculate Half the Difference of Angles

Now, calculate half the difference of the angles. \[\frac{102^{\circ} - 95^{\circ}}{2} = 3.5^{\circ}\]

Step 4: Apply the Difference of Sines Formula

Substitute these values into the identity: \[\sin 102^{\circ} - \sin 95^{\circ} = 2 \cos(98.5^{\circ}) \sin(3.5^{\circ})\].

Step 5: Verify the Solution

The expression \(2 \cos(98.5^{\circ})\sin(3.5^{\circ})\) is the product form of the given trigonometric expression, achieved using a standard identity and calculations.

Key Concepts

Difference of Sines

The **difference of sines** formula is a special trigonometric identity used to transform the difference between two sine values into a product of cosine and sine functions. This transformation simplifies complex trigonometric expressions, allowing easier calculation and understanding. The formula is:\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right) \]

• A and B represent the angles involved in the sine functions.
• The formula expresses the difference as a product of a cosine function, which averages the angles, and a sine function, considering their half difference.

Applying this identity helps break down expressions involving sine functions into a form that can be easily computed or further simplified for practical applications.
For example, in the problem where we need to simplify \( \sin 102^{\circ} - \sin 95^{\circ} \), we use this identity to make the expression manageable by transforming it into a product of trigonometric functions.

Cosine Function

The **cosine function** is a fundamental trigonometric function denoted as \(\cos\), and it describes the ratio of the length of the adjacent side of a right triangle to the hypotenuse, given an angle. It is one of the core elements used in trigonometric identities, including the difference of sines formula, because it provides the "average" value needed in such transformations.

• The angle used in the cosine function in the difference of sines identity comes from the average of the angles \(A\) and \(B\).
• Specifically, it is calculated as \(\frac{A+B}{2}\).
• This angle gives a central value that reflects the midpoint behavior of the trigonometric expression being simplified.

Cosine values are periodic and repeat every \(360^{\circ}\), making them invaluable in calculating periodic phenomena. In our previous example, \(\cos(98.5^{\circ})\) gives the cosine value of the average angle, allowing the trigonometric identity to transform the difference into a simpler product form.

Sine Function

The **sine function** is another vital trigonometric function which is symbolized as \(\sin\). It represents the ratio of the length of the side opposite a given angle to the hypotenuse in a right triangle. The sine function is useful for transforming expressions involving trigonometric identities.

• In the difference of sines formula, the sine function corresponds to the half-difference of the angles \(A\) and \(B\).
• It specifically uses \(\frac{A-B}{2}\) to express the dynamic variation of the angles as a product.
• This aspect of the identity focuses on how the angle difference contributes to the overall behavior of the trigonometric expression.

The function's periodic nature, repeating every \(360^{\circ}\), allows it to variably represent different phases of wave-like phenomena. For the expression involving \(\sin 102^{\circ} - \sin 95^{\circ}\), using \(\sin(3.5^{\circ})\) connects the finer changes in angle directly into the product form, simplifying the problem by focusing on incremental changes in the function.