Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \tan 3 u=\frac{\tan u\left(3-\tan ^{2} u\right)}{1-3 \tan ^{2} u} $$

Step-by-Step Solution

Answer

The identity is verified because both sides simplify to the equation of \( \tan 3u = \frac{3\tan u - \tan^3 u}{1 - 3\tan^2 u} \).

Step 1: Recall the Triple Angle Formula

The given identity involves the tangent of triple an angle. Recall the formula for the tangent of a triple angle: \[ \tan 3u = \frac{3\tan u - \tan^3 u}{1 - 3\tan^2 u} \]. We will use this known identity to verify the given expression.

Step 2: Simplify the Right Side of the Identity

The right-hand side of the identity is \( \frac{\tan u\left(3-\tan ^{2} u\right)}{1-3 \tan ^{2} u} \). Simplify this expression by using the distributive property for the numerator: \[ 3\tan u - \tan^3 u \]. This matches the numerator of the triple angle tangent formula.

Step 3: Compare with the Triple Angle Formula

Now, compare the simplified right-hand side expression \( \frac{3\tan u - \tan^3 u}{1 - 3\tan^2 u} \) with the known formula for \( \tan 3u \), which is exactly the same: \[ \tan 3u = \frac{3\tan u - \tan^3 u}{1 - 3\tan^2 u} \]. The expression matches, thus verifying the identity.

Key Concepts

Triple Angle Formula

The triple angle formula is a crucial element in trigonometry that relates the tangent of an angle multiplied by three to more manageable expressions. For tangent, the triple angle formula is given by: \[ \tan 3u = \frac{3\tan u - \tan^3 u}{1 - 3\tan^2 u} \] This formula is derived from the general idea of angle addition and is an integral part of simplifying complex trigonometric expressions. By using this identity, one can evaluate or verify expressions involving the tangent of three times an angle with greater ease. The formula provides a straightforward way to transform calculations that might otherwise seem intricate or cumbersome. Utilization of this identity can help break down trigonometric identities into simpler, more digestible parts, making mathematical problems less intimidating.

Tangent Function

The tangent function, often denoted as \( \tan \), is one of the principal functions in trigonometry. For an angle \( u \), it is defined as the ratio of the sine of \( u \) to the cosine of \( u \), i.e., \[ \tan u = \frac{\sin u}{\cos u} \] It plays a key role in various branches of mathematics and science. The tangent function is known for its unique properties, including periodicity, where \( \tan(u + \pi) = \tan u \). This behavior makes it predictable over regular intervals and useful for solving problems related to periodic phenomena. Due to its ratio-based definition, the tangent function can easily grow large or become undefined when approaching angles where \( \cos u = 0 \). Nevertheless, understanding and employing the tangent function is essential for mastering trigonometric identities and solving complex mathematical queries.

Identity Verification

Identity verification in trigonometry involves checking if two expressions, often equated, are indeed equal for all values within their domain. Whether verifying the identity of a single formula or proving the equivalence of different expressions, it requires deep knowledge of trigonometric properties and formulas. For verifying an identity, follow these steps:

• Start by examining both sides of the given equation separately.
• Simplify as much as possible using known algebraic and trigonometric identities.
• Adjust either side to match the other by applying transformations like the distributive property or factoring.
• Conclude when both simplified forms reflect identical statements.

In our example, this approach confirms that \[ \tan 3u = \frac{\tan u (3-\tan ^{2} u)}{1-3 \tan ^{2} u} \] is valid by showing it simplifies to the known triple angle formula. Mastering identity verification is crucial as it empowers students to verify and validate complex trigonometric expressions with confidence.