Problem 81 - Precalculus with Limits

Question

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \sin \dfrac{\pi}{3} \cos \dfrac{\pi}{6} \)

Step-by-Step Solution

Answer

The product \( \sin \dfrac{\pi}{3} \cos \dfrac{\pi}{6} \) when converted to a sum using the product-to-sum formula in trigonometry, equals \( \frac{1}{2} \).

Step 1: Identify the appropriate formula

Recognize that the product to sum formula that needs to be used here is \( \sin a \cos b = \frac{1}{2} [\sin(a+b) + \sin (a-b)] \)

Step 2: Plugging the values into the formula

Substitute \( a = \dfrac{\pi}{3} \) and \( b = \dfrac{\pi}{6} \) into the formula. This gives us: \( \sin \dfrac{\pi}{3} \cos \dfrac{\pi}{6} = \frac{1}{2} [\sin(\dfrac{\pi}{3} + \dfrac{\pi}{6}) + \sin (\dfrac{\pi}{3} - \dfrac{\pi}{6})] \)

Step 3: Calculating the sum

Calculate the sum in the equation. This simplifies to: \( \frac{1}{2} [\sin(\dfrac{5\pi}{6}) + \sin (\dfrac{\pi}{6})] \)

Step 4: Evaluating and Simplifying

Evaluate the sine values: \( \frac{1}{2} [\frac{1}{2} + \frac{1}{2}] = \frac{1}{2} \)

Key Concepts

Trigonometric Identities

Trigonometric identities are foundational elements in the study of trigonometry, providing essential relationships between the trigonometric functions. These identities are true for all values of the variables. A fundamental category of these identities is the product-to-sum formulas, which enable the simplification of the product of sine and cosine functions into a sum or difference of sines or cosines.

For instance, the product-to-sum identity \( \sin a \cos b = \frac{1}{2} [\sin(a+b) + \sin (a-b)] \) can be utilized to rewrite the expression \( \sin \dfrac{\pi}{3} \cos \dfrac{\pi}{6} \) as used in the exercise. Understanding these identities helps in transforming complex trigonometric expressions into more manageable forms, which is especially beneficial for integrating or solving trigonometry problems.

Sine and Cosine Functions

Sine and cosine are the fundamental trigonometric functions, relating the angles of a triangle to the lengths of its sides in a right-angled triangle. These functions are periodic and oscillate between -1 and 1. The sine function (sin) measures the ratio of the opposite side to the hypotenuse, while the cosine function (cos) measures the ratio of the adjacent side to the hypotenuse.

When dealing with angles in radians, such as \( \dfrac{\pi}{3} \) and \( \dfrac{\pi}{6} \) in our textbook exercise, knowing the exact value of sine and cosine for standard angles can simplify calculations considerably. Remembering that \( \sin \dfrac{\pi}{6} = \frac{1}{2} \) and \( \cos \dfrac{\pi}{6} = \frac{\sqrt{3}}{2} \) allows us to substitute these values directly into the product-to-sum formula.

Trigonometry in Precalculus

Trigonometry is an essential part of precalculus, setting the stage for advanced studies in mathematics. It encompasses the study of the properties of triangles, the relationships between side lengths and angles, and the applications of trigonometric functions and identities.

In precalculus, students learn how to manipulate and graph these functions, solve trigonometric equations, and apply identities to complex mathematical problems. Trigonometry is not only theoretical but also has practical applications in fields like engineering, physics, and astronomy. Mastering concepts like the product-to-sum formulas in trigonometry equips students with problem-solving skills necessary for calculus and beyond.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a common task in trigonometry which involves reducing expressions to a more basic or easily comprehensible form. To do this, one typically uses trigonometric identities, like the product-to-sum formulas, to transform products into sums or differences, or vice versa.

This process is not only about making the expression look easier, but it also prepares the expression for further operations such as integration, differentiation, or solving equations. As seen in the exercise solution, simplifying \( \sin \dfrac{\pi}{3} \cos \dfrac{\pi}{6} \) using a product-to-sum formula turned the product of sine and cosine into a form that could be evaluated directly, streamlining the calculation process significantly.