Problem 51 - Precalculus Essentials

Question

Verify each identity. \(\sin 2 \alpha=2 \sin \alpha \cos \alpha\) Hint: Write \(\sin 2 \alpha\) as \(\sin (\alpha+\alpha)\)

Step-by-Step Solution

Answer

The identity \(\sin 2 \alpha=2 \sin \alpha \cos \alpha\) is verified by applying the sum of angles identity for sine to the left side of the equation, based on the hint given in the exercise, then simplifying the equation.

Step 1: Expanding the Left Side with Sum Formula

First, rewrite the left side of the equation, \(\sin 2 \alpha\), by expressing 2 alpha as the sum of alpha and alpha. This results in \(\sin(\alpha + \alpha)\). Then apply the sum of angles identity for sine, \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) with a = b = alpha. Substituting, we get \(\sin(\alpha + \alpha) = \sin \alpha \cos \alpha + \cos \alpha \sin \alpha\).

Step 2: Simplifying the Equation

Next, notice that \(\sin \alpha \cos \alpha + \cos \alpha \sin \alpha\) are identical terms, so they can be combined into 2\( \sin \alpha \cos \alpha\). This implies that \( \sin(\alpha + \alpha) = 2 \sin \alpha \cos \alpha \).

Step 3: Verifying the Identity

Finally, notice that \(\sin(\alpha + \alpha) = 2 \sin \alpha \cos \alpha \) is exactly the same as the given identity \(\sin 2 \alpha=2 \sin \alpha \cos \alpha\). Hence, the identity has been verified.

Key Concepts

Sum of Angles Identity

When studying trigonometric identities, one essential tool is the sum of angles identity. It helps in breaking down complex trigonometric expressions into simpler ones that are easier to handle. Specifically, for the sine function, the sum of angles identity states:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

This identity is powerful because it allows us to express the sine of a sum of two angles in terms of the sines and cosines of the individual angles.To apply this identity, consider \( \sin(2\alpha) \). We rewrite \( 2\alpha \) as \( \alpha + \alpha \), so we can use the sum of angles identity. By replacing \( a \) and \( b \) with \( \alpha \) and \( \alpha \) respectively, we can simplify or solve various expressions or identities. Understanding this identity is crucial for verifying trigonometric identities effectively.

Sine Function

The sine function is a fundamental part of trigonometry with its roots in the geometry of right triangles and the unit circle.Here are some key aspects:

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
• On the unit circle, sine represents the y-coordinate of a point corresponding to a given angle.

For example, when verifying the identity \( \sin 2\alpha = 2\sin \alpha \cos \alpha \), this involves connecting the concepts of sine and cosine. Understanding \( \sin \) in terms of both angles and their geometric meaning allows deeper comprehension and application of such identities. This is especially useful for confirming that seemingly complicated expressions equal each other using known trigonometric relationships.

Verification of Identities

Verification of trigonometric identities often involves rewriting one side of an equation to show it is equivalent to the other.This requires familiarity with various trigonometric formulas and properties. Here’s how you typically approach it:

• Identify which identity or formula best fits the situation.
• Apply the identity to transform the expression.
• Simplify it through algebraic manipulation if necessary.
• Check both sides of the equation to ensure they match.

For example, to verify \( \sin 2\alpha = 2\sin \alpha \cos \alpha \), we start by applying the sum of angles identity: rewinding \( \sin(\alpha + \alpha) \) to \( \sin \alpha \cos \alpha + \cos \alpha \sin \alpha \). Then, we simplify to get \( 2\sin \alpha \cos \alpha \), matching the right side. Successfully verifying an identity gives confidence in understanding trigonometric relationships and their applications.