Problem 39 - Precalculus Essentials

Question

Use words to describe the given formula. $$\sin \alpha \sin \beta=\frac{1}{2}[\cos (\alpha-\beta)-\cos (\alpha+\beta)]$$

Step-by-Step Solution

Answer

The formula is saying that the product of the sin of α and the sin of β equals one half of the difference of the cosine of (α - β) and the cosine of (α + β).

Step 1: Interpret the given formula

The given formula is as follows: the product of sin of α and sin of β is equal to half of the difference of cosine of (α - β) and cosine of (α + β).

Step 2: Explaining Trigonometric Functions Involved

1. \( \sin \alpha \) and \( \sin \beta \) are the sine functions of angle α and angle β respectively. 2. \( \cos (\alpha - \beta) \) and \( \cos (\alpha+\beta) \) are cosine functions of angle α minus angle β and angle α plus angle β respectively.

Step 3: Explaining Equality and Operations

The product or multiplication of \( \sin \alpha \) and \( \sin \beta \) is equal to one-half the subtraction of \( \cos (\alpha - \beta) \) from \( \cos (\alpha+\beta) \).

Key Concepts

Sine Function

The sine function is one of the fundamental concepts in trigonometry. It relates the ratio of the lengths of the opposite side to the hypotenuse in a right-angled triangle. The sine of an angle, often denoted as \( \sin \theta \), serves as a way to measure the angle using these lengths.

Some key points about the sine function include:

• It is periodic with a period of \( 2\pi \).
• \( \sin \theta \) is positive in the first and second quadrants, and negative in the third and fourth quadrants.
• The range of the sine function is from -1 to 1 inclusive.
• At \( \theta = 0, \sin \theta = 0 \), at \( \theta = \frac{\pi}{2}, \sin \theta = 1 \), and \( \theta = \pi, \sin \theta = 0 \) again, exhibiting its periodicity.

Understanding the sine function is crucial for solving problems involving angles and their relationships in trigonometry.

Cosine Function

Similar to the sine function, the cosine function is essential in trigonometry. The cosine of an angle relates the adjacent side's length to the hypotenuse in a right-angled triangle. Denoted as \( \cos \theta \), the cosine function offers another perspective for measuring angles.

Key characteristics of the cosine function:

• It shares the same periodicity as the sine function with a period of \( 2\pi \).
• \( \cos \theta \) is positive in the first and fourth quadrants, and negative in the second and third quadrants.
• Also ranging from -1 to 1, it represents maximal values at \( \theta = 0 \) with \( \cos \theta = 1 \), and minimal values at \( \theta = \pi \) where \( \cos \theta = -1 \).

Understanding both sine and cosine functions and their periodic nature helps solve more complex trigonometric identities, like using them in angle difference formulas.

Angle Difference Formulas

Angle difference formulas are pivotal in simplifying complex trigonometric expressions. They help by expressing trigonometric functions of compound angles in terms of the functions of individual angles.

When dealing with expressions such as \( \cos(\alpha - \beta) \) or \( \sin(\alpha + \beta) \), angle difference and sum formulas become very useful. For the cosine function:

• \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \)
• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

For the sine function:

• \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)
• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)

Utilizing these formulas provides a clear pathway to express complex trigonometric products and sums into simpler forms, as seen in exercises involving identifications of formulas like \( \sin \alpha \sin \beta = \frac{1}{2}[\cos (\alpha-\beta) - \cos (\alpha+\beta)] \).