Problem 1 - Precalculus

Question

In Exercises \(1-6,\) use the Even / Odd Identities to verify the identity. Assume all quantities are defined. $$ \sin (3 \pi-2 \theta)=-\sin (2 \theta-3 \pi) $$

Step-by-Step Solution

Answer

The identity is verified using the odd sine function identity: \( \sin(x) = -\sin(-x) \).

Step 1: Use the Identity for Sine

Recall that the sine function has the property that \( \sin(-x) = -\sin(x) \). We will use this identity to simplify \( \sin(3\pi - 2\theta) \).

Step 2: Apply the Transformation

Note that \( 3\pi - 2\theta = -(2\theta - 3\pi) \). So \( \sin(3\pi - 2\theta) = \sin(-(2\theta - 3\pi)) \).

Step 3: Apply the Odd Identity for Sine

Using the identity \( \sin(-x) = -\sin(x) \), it follows that \( \sin(3\pi - 2\theta) = -\sin(2\theta - 3\pi) \).

Step 4: Verify the Identity

Now we have shown that \( \sin(3\pi - 2\theta) = -\sin(2\theta - 3\pi) \), confirming the identity given in the problem.

Key Concepts

Even and Odd Functions

Even and odd functions are a part of the fundamental concepts of trigonometry. Understanding these concepts can simplify complex problems and is essential for verifying identities. Functions are classified based on their symmetry:

• Even Functions: These have symmetry about the y-axis, meaning the function satisfies the condition \( f(-x) = f(x) \). This symmetry implies that the graph of an even function remains unchanged when reflected across the y-axis. A classic example of an even function is the cosine function, \( \cos(x) \), which satisfies \( \cos(-x) = \cos(x) \).
• Odd Functions: These have rotational symmetry about the origin, which means the function satisfies \( f(-x) = -f(x) \). Odd functions change sign when their input is negated. A key example of an odd function is the sine function, \( \sin(x) \), which follows \( \sin(-x) = -\sin(x) \).

Understanding these properties can help you recognize patterns and simplify expressions, which was crucial in solving the problem by verifying the equality of two sine expressions.

Sine Function

The sine function is one of the fundamental trigonometric functions and is crucial in both algebra and geometry. Its behavior as an odd function sets it apart:

• Definition: The sine function, represented as \( \sin(x) \), gives the y-coordinate of a point on the unit circle corresponding to an angle \( x \), measured in radians.
• Periodic Nature: The sine function is periodic with a period of \( 2\pi \). This means \( \sin(x + 2\pi) = \sin(x) \), reflecting its cyclical behavior.
• Odd Function Property: As an odd function, \( \sin(-x) = -\sin(x) \). This property is pivotal when dealing with transformations involving negative angles, as it allows you to simplify expressions by factoring out negative signs.

In the given exercise, the odd nature of the sine function helped verify the identity by recognizing that a sign change in the input directly affects the output, confirming the relationship between the angles given.

Angle Transformation

Angle transformations involve manipulating angles to achieve equivalent expressions or desired forms.

• Understanding Transformation: Consider the transformation \( 3\pi - 2\theta \), which could be rewritten using the identity \( 3\pi - 2\theta = -(2\theta - 3\pi) \). This shows that the two angles are negatives of each other.
• Application in Trigonometry: Such transformations often simplify the process of solving trigonometric equations or verifying identities. Recognizing angle equivalences or complementary angles can transform complex expressions into simpler forms.
• Link with Odd Functions: In the context of the sine function, transformations involving negative angles make the odd function property applicable, allowing further simplification by recognizing symmetry properties.

In the exercise, angle transformation helped set the stage for applying the odd identity for sine, ultimately leading to the verification of the provided trigonometric identity.