Problem 27 - Precalculus: Functions and Graphs

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 by matching it to the known triple angle formula for tangent.

Step 1: Recall the Triple Angle Identity for Tangent

The triple angle identity for tangent is given by: \( \tan(3u) = \frac{3\tan(u) - \tan^3(u)}{1 - 3\tan^2(u)} \). This identity will help us verify the given statement.

Step 2: Express the Numerator using Tangent Identity

Notice the given expression: \( \frac{\tan u (3 - \tan^2 u)}{1 - 3 \tan^2 u} \). Start with the numerator: \( \tan u (3 - \tan^2 u) = 3\tan u - \tan^3 u \). This matches the numerator in the triple angle identity for tangent.

Step 3: Compare Denominators

The denominator in the given expression is \( 1 - 3 \tan^2 u \). This matches the denominator in the triple angle identity for \( \tan(3u) = \frac{3\tan(u) - \tan^3(u)}{1 - 3\tan^2(u)} \).

Step 4: Conclusion on Verification

Since both the numerator \( 3\tan u - \tan^3 u \) and the denominator \( 1 - 3\tan^2 u \) of the given expression match the triple angle identity for tangent \( \tan(3u) \), the identity \( \tan 3u = \frac{\tan u (3 - \tan^2 u)}{1 - 3 \tan^2 u} \) is verified.

Key Concepts

Triple Angle Identity

The Triple Angle Identity is a fundamental concept in trigonometry. It relates the angle-at-three-times of a trigonometric function to the angle itself. For the tangent function, the relationship is expressed as: \[ \tan(3u) = \frac{3\tan(u) - \tan^3(u)}{1 - 3\tan^2(u)} \] This identity helps simplify the process of dealing with trigonometric functions at multiplied angles.
By understanding and utilizing this identity, we can solve trigonometric equations more efficiently. For example, if we need to find \(\tan(3u)\), instead of dealing with \(3u\) directly, we use this formula with \(u\). This is especially useful when \(u\) represents an angle in mathematical problems.
Knowing this identity allows us to verify complex function expressions, make calculations simpler, and understand trigonometric relations on a deeper level.

Tangent Function

The tangent function is one of the primary functions in trigonometry. It is defined as the ratio of the sine to the cosine of an angle, given by:
\[ \tan(u) = \frac{\sin(u)}{\cos(u)} \] This function is periodic, with a period of \(\pi\), meaning it repeats every \(\pi\) radians.
The tangent function is unique in that it can produce values ranging from negative infinity to positive infinity. Unlike sine and cosine, which are limited to values between -1 and 1, tangent can deal with vertical asymptotes due to its undefined nature at angles where the cosine is zero.
The behavior of the tangent function makes it important in geometry and physics, especially when calculating angles and distances. It can be manipulated using identities like the triple angle identity to solve equations or verify expressions, as seen in trigonometric verification processes.

Trigonometric Verification

Trigonometric verification is a process used to prove that two trigonometric expressions are equivalent. This is crucial in many mathematical applications, from solving equations to simplifying complex expressions.
To verify the identity \( \tan(3u) = \frac{\tan u (3 - \tan^2 u)}{1 - 3 \tan^2 u} \), we compare each part of both sides of the equation. We use known identities, such as the triple angle identity, for this process.

• First, ensure both the numerator and the denominator of the given expression align with the components of the identity.
• Check if the transformed parts of the expression on one side correspond exactly to the identity recognised in the known formulae.
• Determining this match means the expressions are indeed equivalent, completing the verification process.

Engaging with trigonometric verification not only reinforces an understanding of identities and functions but also enhances problem-solving skills that are vital in advanced mathematics.