Problem 84 - Precalculus with Limits

Question

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( 6 \sin 45^\circ \cos 15^\circ \)

Step-by-Step Solution

Answer

The simplified expression is \(3[\sin(60^\circ) + \sin(30^\circ)]\).

Step 1: Identifying the Formula to Use

Use the product-to-sum formula in the form: \(\sin(a)\cos(b) = \frac{1}{2}[\sin(a+b) + \sin(a-b)]\) for this exercise.

Step 2: Substitute the Trigonometric Values

Substitute \(a = 45^\circ\) and \(b = 15^\circ\) into the formula: \(6 \cdot \frac{1}{2}[\sin(45^\circ+15^\circ) + \sin(45^\circ-15^\circ)]\).

Step 3: Simplification

Simplify the expression: 3[\sin(60^\circ) + \sin(30^\circ)].

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They play a crucial role in simplifying expressions and solving trigonometry problems. One of the most important sets of trigonometric identities is the product-to-sum formulas. These formulas allow us to express products of trigonometric functions as sums or differences, which can simplify complex trigonometry problems.

An example of a product-to-sum formula is \(\sin(a)\cos(b) = \frac{1}{2}[\sin(a+b) + \sin(a-b)]\). In our exercise, we utilized this specific identity to convert the product of \(\sin 45^\circ\) and \(\cos 15^\circ\) into a sum. Remember, knowing these identities and how to apply them is essential for any student wishing to excel in trigonometry.

Trigonometric Functions

Trigonometric functions, such as sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), are functions of an angle and are foundational to the study of trigonometry. These functions describe ratios of sides in right-angled triangles and can also be represented as points on the unit circle.

For example, the sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse. In the context of our example, we are dealing with \(\sin 45^\circ\) and \(\cos 15^\circ\), where 45 and 15 degrees are our angles of interest, and their sine and cosine values can be found using the unit circle or trigonometric tables. Understanding how these functions work and relate to angles is indispensable for solving trigonometric equations and problems.

Trigonometry Problem Solving

Trigonometry problem solving requires a combination of analytical skills and a strong grasp of trigonometric concepts. The process typically follows identifying the right trigonometric identities or formulas to apply, substituting the given values or variables, and then simplifying the mathematical expression to find the answer.

In our exercise, we started by recognizing the form of the product we were dealing with (\(\sin\) times \(\cos\)) and then selected the correct product-to-sum formula. Substitution involved plugging the given angles into the formula, and simplification encompassed performing the arithmetic operations to arrive at the final simplified sum. This step-by-step approach, along with practice and familiarity with fundamental trigonometric functions and identities, enables students to tackle even the most daunting trigonometry problems.