Problem 74 - Algebra and Trigonometry

Question

Rewrite each expression as a simplified expression containing one term. $$\sin \left(\frac{\pi}{3}-\alpha\right) \cos \left(\frac{\pi}{3}+\alpha\right)+\cos \left(\frac{\pi}{3}-\alpha\right) \sin \left(\frac{\pi}{3}+\alpha\right)$$ (Do not use four different identities to solve this exercise.)

Step-by-Step Solution

Answer

The simplified expression is \(\sin\left(\frac{2\pi}{3}\right)\).

Step 1: Identify the form of the given expression

Looking at the original expression, it appears to be a sum of products of sine and cosine functions. Each product involves a subtraction and an addition of the same two terms. This is in the form of \(\sin\left(a-b\right) \cos\left(a+b\right)+\cos\left(a-b\right) \sin\left(a+b\right)\).

Step 2: Use the correct formula

The next step is to simplify this expression by using the appropriate identities. The formula used here is: \(\sin(a) \cos(b) + \cos(a) \sin(b) = \sin(a + b)\). So, substituting the parameters, the expression becomes: \(\sin\left(\frac{\pi}{3} - \alpha + \frac{\pi}{3} + \alpha\right)\).

Step 3: Simplify the expression

We then simplify the expression by cancelling out the \(- \alpha\) and \(+ \alpha\) to get the final simplified term. The expression will become: \(\sin\left(\frac{\pi}{3} + \frac{\pi}{3}\right) = \sin\left(\frac{2\pi}{3}\right)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are fundamental in simplifying complex trigonometric expressions and solving trigonometry problems. Some of the most common trigonometric identities include the Pythagorean identities, angle sum and difference identities, and the double angle identities. These identities are powerful tools as they transform trigonometric functions into alternative forms that can make calculations simpler and more intuitive. Understanding how and when to apply these identities is crucial for students who are aiming to master trigonometry problem solving.

For example, in the given exercise, the sum-to-product identity is essential to combine two trigonometric terms into a single function. This demonstrates the utility of having a grasp over various trigonometric identities to simplify expressions effectively.

Sum-to-Product Identity

The sum-to-product identity is used to convert a sum or difference of two trigonometric functions into a product of sines and cosines. This can make further calculation more manageable, especially when dealing with integrals or other complex operations. The identity employed in the exercise is one of these, specifically the sine of sum identity: \( \sin(a) \cos(b) + \cos(a) \sin(b) = \sin(a + b) \).

By applying this identity, students can rewrite the expression in a more condensed form that is easier to evaluate or integrate, if necessary. It's important to not only memorize these identities but also understand how they are derived, as this deepens comprehension and facilitates their correct application in various problems.

Sine and Cosine Functions

Sine and cosine are the fundamental trigonometric functions and are the building blocks for all other trigonometric functions. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Similarly, the cosine is the ratio of the length of the adjacent side to the hypotenuse.

Both functions have a range of \(-1\) to \(1\) and repeat every \(2\pi\) radians, known as the period. They are also functions of an angle, which can represent either a rotation or a triangle in trigonometry. Understanding the sine and cosine functions, their graphs, and their properties is critical when analyzing and simplifying trigonometric expressions.

Trigonometry Problem Solving

Solving trigonometry problems often involves identifying the type of problem, selecting the appropriate trigonometric identities, and applying algebraic manipulations. Strong problem-solving skills in trigonometry include the ability to recognize patterns, understand the properties of trigonometric functions, and a thorough familiarity with identities such as angle sum and difference, double angle, and sum-to-product identities.

In practice, as illustrated in the given exercise, problem solving begins with simplifying the expression using identities to make the equation more accessible. Problem solvers must also be careful not to introduce errors during simplification, such as neglecting to consider the domain of the functions involved. By mastering these techniques, students can tackle a wide range of trigonometry problems with confidence.