Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \sin 3 u=\sin u\left(3-4 \sin ^{2} u\right) $$

Step-by-Step Solution

Answer

The identity is verified using the triple angle formula for sine.

Step 1: Recall the triple angle formula for sine

The triple angle formula for sine is given by \( \sin 3u = 3 \sin u - 4 \sin^3 u \). Use this identity to verify the given equation.

Step 2: Compare the given identity with the triple angle formula

The given identity is \( \sin 3u = \sin u (3 - 4 \sin^2 u) \). Notice that this is in the form \( a = b \). We need to confirm that both sides of the equation are equal using known identities.

Step 3: Expand the right-hand side of the given identity

Expand \( \sin u (3 - 4 \sin^2 u) \) using distribution: \( 3 \sin u - 4 \sin^3 u \).

Step 4: Compare the expanded form with the triple angle formula

Both the expanded form \( 3 \sin u - 4 \sin^3 u \) and the triple angle formula \( 3 \sin u - 4 \sin^3 u \) match exactly, thus verifying the identity.

Key Concepts

Triple Angle Formula

Understanding the triple angle formula is key to solving trigonometric identities involving multiple angles. The triple angle formulas help you express trigonometric functions of angles like \(3u\) in terms of trigonometric functions of \(u\) itself.
For the sine function, the triple angle formula is:

• \(\sin 3u = 3\sin u - 4\sin^3 u\)

This formula is incredibly useful because it breaks down a more complicated angle problem into simpler parts by involving powers of single sine functions.
The component \(3\sin u\) shows a direct linear relation, while \(-4\sin^3 u\) introduces a cubic term that often appears in trigonometric identity proofs.
Remember that this formula derives from adding angle identities repeatedly, so a solid grasp of addition formulas for sine will also be beneficial.

Sine Function

The sine function is one of the fundamental trigonometric functions, usually represented by \(\sin x\), where \(x\) is an angle. It's defined in the context of a right triangle as the ratio of the opposite side to the hypotenuse, but it also extends to describe a continuous wave pattern over all real numbers using the unit circle.
On the unit circle, \(\sin x\) corresponds to the y-coordinate of a point as it travels around the circle. This property allows us to easily connect sine with other trigonometric identities and angle measures, even beyond just triangles.
In using sine to verify identities such as the triple sine formula, keep in mind:

• The function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) units.
• It has a range of \([-1, 1]\), thus values will always fall within this interval, crucial when comparing both sides of an identity.

These properties help greatly in understanding how sine behaves in various identities and verifying them effectively.

Identity Verification

Identity verification in trigonometry involves proving that two different expressions are equivalent. It requires a methodical approach and often uses known identities to transform one or both sides of the equation until they match.
To verify the given identity \(\sin 3u = \sin u(3 - 4 \sin^2 u)\), follow these basic steps:

• Identify which known identities might be useful. For this exercise, we used the triple angle formula for sine.
• Simplify or transform one side of the equation using these identities. Here, we expanded the right-hand side \(\sin u(3 - 4 \sin^2 u)\) to get \(3 \sin u - 4 \sin^3 u\).
• Ensure both sides of the equation. If they match, the identity is verified; if not, re-evaluate your steps.

This process underpins much of trigonometric identity verification, encouraging precision and an understanding of trigonometric relationships.