Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin 5 \theta-\sin \theta$$

Step-by-Step Solution

Answer

The expression \(\sin 5\theta - \sin \theta\) simplifies to \(2 \cos(3\theta) \sin(2\theta)\) using the sum-to-product identities.

Step 1: Identifying A and B

In the given expression, \(A = 5\theta\) and \(B = \theta\). The goal is to transform the expression into the form of the identity.

Step 2: Applying the Identity

Plug the values of A and B into the identity: \(2 \cos \left(\frac{(5\theta + \theta)}{2}\right) \sin \left(\frac{(5\theta - \theta)}{2}\right)\)

Step 3: Simplifying the Expression

Simplify the expression by doing the calculations inside the parentheses: \(2 \cos(3\theta) \sin(2\theta)\)

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in solving trigonometry problems. They are equations that involve trigonometric functions and are true for any value plugged into them. Sum-to-product formulas are a specific type of trigonometric identity that allow us to write the sum or difference of sine and/or cosine functions as a product of sines and cosines. The formula used in the exercise is one such identity, which states that for any angles A and B:

\[ \sin(A) - \sin(B) = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]

Using this identity simplifies complex expressions and enables further algebraic manipulation. Understanding these identities is crucial as it provides a foundational structure for working through more advanced trigonometry problems.

Sine Function

The sine function is one of the basic trigonometric functions, which relates a real number t to the y-coordinate of the point on the unit circle that is reached by moving t units along the circumference, starting from the rightmost point of the circle (1,0). For any angle \( \theta \), the sine function is defined as:

\[ \sin(\theta) = \text{y-coordinate of the point on the unit circle} \]

The sine function is periodic with a period of \(2\pi\) radians or 360 degrees, meaning \( \sin(\theta) = \sin(\theta + 2\pi k) \) for any integer k. In the context of our exercise, the sine function is manipulated using the sum-to-product formula to express the sum of two sine functions as a product.

Algebraic Manipulation

Algebraic manipulation involves the use of algebraic rules to rearrange and simplify expressions. It is an essential skill in mathematics that allows for the transformation of complex equations into simpler forms that are easier to understand and solve.

Steps in Algebraic Manipulation

• Identify the components of the expression: In our exercise, \(A = 5\theta\) and \(B = \theta\) were identified in the sum.
• Use appropriate formulas: The sum-to-product formula was applied to the sine functions in the expression.
• Simplify the result: Perform arithmetic operations to get a simplified expression, which, in our case, yielded \(2 \cos(3\theta) \sin(2\theta)\).

Algebraic manipulation is not just about following rules, but also about understanding the underlying concepts to apply them effectively. This is evident in how we transformed the sum of sines into a product, a strategy often employed in trigonometry problem solving.

Trigonometry Problem Solving

Trigonometry problem solving involves identifying the right approach to simplify and solve problems involving angles and lengths in right-angled triangles or on the unit circle. It encompasses a combination of strategies such as:

• Recognizing patterns and selecting appropriate trigonometric identities.
• Algebraic manipulation to simplify expressions.
• Geometrical understanding of trigonometric functions.
• Applying the properties of angles, such as complementary and supplementary angles.

The sum-to-product transformation seen in the textbook exercise is a great example of trigonometry problem solving. Turning a sum into a product often simplifies the integration or differentiation of trigonometric expressions, thus making it an invaluable technique in advanced mathematics, including calculus.