Question

Show \(\sin 3 A=3 \sin A \cos ^{2} A-\sin ^{3} A\)

Short answer

Question: Prove the trigonometric identity \(\sin 3 A = 3 \sin A \cos^2 A - \sin^3 A\). Answer: Following four steps, we first applied the sine sum identity to \(\sin 3 A\), then used double-angle identities for \(\sin 2 A\) and \(\cos 2 A\), distributed and simplified the expression, and finally rearranged the terms to match the given identity. Thus, we proved that \(\sin 3 A = 3 \sin A \cos^2 A - \sin^3 A\).

Step-by-step solution

Apply the sine sum identity to \(\sin 3 A\)

We want to find the expression of \(\sin 3 A\). To do this, we will use the sine sum identity: \(\sin(x + y) = \sin x \cos y + \cos x \sin y\). We can rewrite \(\sin 3 A\) as \(\sin(2 A + A)\), then apply the sine sum identity: \(\sin 3 A = \sin(2 A + A) = \sin 2 A \cos A + \cos 2 A \sin A\).

Use double-angle identities for \(\sin 2 A\) and \(\cos 2 A\)

Now, we need to simplify the expression by using the double-angle identities for sine and cosine: \(\sin 2x = 2 \sin x \cos x\) and \(\cos 2x = 1 - 2\sin^2 x\). Replace these identities in the expression from Step 1: \(\sin 3 A = (2 \sin A \cos A) \cos A + (1 - 2\sin^2 A) \sin A\)

Distribute and simplify the expression

Distribute the terms and simplify the resulting expression: \(\sin 3 A = 2 \sin A \cos^2 A + \sin A - 2\sin^3 A\)

Rearrange the terms to match the given identity

Finally, let's rearrange the terms to match the given identity: \(\sin 3 A = 3\sin A \cos^2 A - \sin^3 A\) Thus, we have shown that \(\sin 3 A = 3 \sin A \cos^2 A - \sin^3 A\) as required.

Key concepts

Sine Sum Identity

When dealing with trigonometric equations, knowing how to simplify expressions using identities is crucial. Among these, the sine sum identity is particularly useful.

It states that for any two angles, say \(x\) and \(y\), the sine of their sum can be expressed as: \[ \sin(x + y) = \sin x \cos y + \cos x \sin y \.\] In the context of the exercise, we apply the sine sum identity to break down \(\sin 3A\) into a more manageable form by considering \(3A\) as \(2A + A\). After applying the identity, we obtain an expression that reveals the relationship between \(\sin 3A\) and the trigonometric functions of \(A\).

Understanding this identity not only assists in this exercise but is also a key to solving a variety of trigonometric problems, making it an essential tool in a student's mathematical arsenal.

Double-Angle Identities

To advance further, we introduce double-angle identities — these allow the simplification of trigonometric expressions where the argument is doubled, such as \(2x\) or \(2A\) in this case.

These identities include:

• For sine: \[ \sin 2x = 2 \sin x \cos x \.\]
• For cosine, there are multiple equivalent forms, but for this exercise we used: \[ \cos 2x = 1 - 2\sin^2 x \.\]

By substituting into the expression obtained from the sine sum identity, we make the problem simpler to handle. This step underlines the importance of double-angle identities in transforming complex trigonometric expressions into ones that are more solvable.

Trigonometric Simplification

The final step in solving the given problem is to simplify the expression obtained after applying double-angle identities. This involves conventional algebraic methods like distributing, combining like terms, and factorization.

In our exercise, after applying these simplification techniques, we reached a form that matches the given identity. Such simplification is a testament to how algebra and trigonometry often come hand in hand to solve problems.

Practicing this type of trigonometric simplification builds foundational skills essential for tackling a wide range of mathematical problems beyond just trigonometry, which often involve intricate manipulations of equations and identities.