Problem 51 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \begin{aligned} &\sin 2 \alpha=2 \sin \alpha \cos \alpha\\\ &\text {Hint: Write } \sin 2 \alpha \text { as } \sin (\alpha+\alpha) \end{aligned} $$

Step-by-Step Solution

Answer

The given identity \(\sin 2 \alpha = 2 \sin \alpha \cos \alpha \) is proven by using sine of sum of two angles identity and algebraic manipulation.

Step 1: Express sin 2α in terms of the sine sum identity.

Write sin 2α as sin(α + α). This indicates that we are dealing with a double angle.

Step 2: Apply sine sum identity.

Replace sin(α + α) with the formula of sine of sum of two angles, which is, sin a cos b + cos a sin b. Therefore, sin(α + α) becomes sin α cos α + cos α sin α.

Step 3: Apply commutative property and simplify.

By applying the commutative property of addition (a + b = b + a), the expression sin α cos α + cos α sin α becomes 2sin α cos α, which means sin 2 α = 2sin α cos α, which verifies the double angle identity for sin.

Key Concepts

Sum and Difference Identities

Understanding the sum and difference identities is critical when working with trigonometric equations and expressions. These identities explain how to find the sine, cosine, and tangent of the sum or difference of two angles using the trigonometric functions of the individual angles.

The sine sum identity, for example, states that for any angles \( \alpha \) and \( \beta \), the sum can be expressed as \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \). The difference identity for sine is similarly structured but uses a minus sign. Understanding these formulas is essential for solving trigonometry problems, as they offer a way to simplify and solve complex trigonometric functions.

Sine Sum Identity

The sine sum identity is particularly useful for solving trigonometric expressions involving the sum of two angles. It allows us to express \( \sin(\alpha + \beta) \) as the sum of products of sines and cosines of the individual angles.

Specifically, the identity reads \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \). This identity is not only crucial for direct calculations but also in proving other trigonometric identities, such as double angle formulas. By setting \( \alpha = \beta \), we can derive the double angle identity for sine, which was demonstrated in the exercise by expressing \( \sin 2\alpha \) as \( \sin(\alpha + \alpha) \).

Commutative Property of Addition

The commutative property of addition is a simple yet foundational concept in mathematics. It states that the order in which two numbers are added does not affect the sum, allowing for \( a + b = b + a \).

In trigonometric identities, this property is frequently applied to rearrange terms for simplification or proof purposes, as seen in the provided exercise. After applying the sine sum identity, we obtained \( \sin\alpha \cos\alpha + \cos\alpha \sin\alpha \). However, by employing the commutative property, we can rewrite this expression as \( 2\sin\alpha \cos\alpha \), which is a more concise and recognizable form of the identity.

Verifying Trigonometric Identities

Verifying trigonometric identities is a process of demonstrating that two trigonometric expressions are equivalent. This verification often involves applying various trigonometric formulas and algebraic manipulations, such as factoring, distributing, and using the commutative or associative properties.

For instance, in the exercise provided, we verified the double angle identity \( \sin 2\alpha = 2\sin\alpha \cos\alpha \) by expressing \( \sin 2\alpha \) as \( \sin(\alpha + \alpha) \), then applying the sine sum identity and utilizing the commutative property of addition for final simplification. Through these steps, we can show the equivalence of the initial and the final expression, thus confirming the identity.