Problem 30 - Precalculus Mathematics for Calculus

Question

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\frac{2 \tan 7^{\circ}}{1-\tan ^{2} 7^{\circ}}\) (b) \(\frac{2 \tan 7 \theta}{1-\tan ^{2} 7 \theta}\)

Step-by-Step Solution

Answer

(a) \(\tan 14^\circ\); (b) \(\tan 14\theta\)

Step 1: Identify the Transformation Formula

Recognize that the expression \(\frac{2 \tan x}{1-\tan^2 x}\) can be rewritten using the tangent double-angle identity: \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\). This formula can help us simplify the given expressions.

Step 2: Apply the Double-Angle Formula for (a)

For (a), set \(x = 7^\circ\), then the expression \(\frac{2 \tan 7^\circ}{1-\tan^2 7^\circ}\) becomes \(\tan 14^\circ\) by using the double-angle identity.

Step 3: Apply the Double-Angle Formula for (b)

For (b), set \(x = 7\theta\), then the expression \(\frac{2 \tan 7\theta}{1-\tan^2 7\theta}\) simplifies to \(\tan 14\theta\) using the same double-angle identity.

Key Concepts

Double-Angle Formula

The double-angle formula is a handy trigonometric identity that helps simplify expressions involving trigonometric functions by doubling the angle. For tangent, the double-angle formula is given by:

• \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \)

This formula comes in particularly useful when needing to convert expressions from one angle measurement to double that angle, enabling simplification of complex trigonometric expressions.
In both examples from the exercise, the expression \(\frac{2 \tan x}{1 - \tan^2 x}\) was identified to match the right side of the tangent double-angle identity. Hence, for example (a), with \(x = 7^\circ\), the expression simplifies directly to \(\tan 14^\circ\). This technique helps remove the tangent operation over complex fractional expressions, making calculations more straightforward.
Similarly, in example (b), where \(x = 7\theta\), the same principle applies, simplifying the expression to \(\tan 14\theta\). Understanding and applying the double-angle formula makes tackling trigonometric problems far easier and more manageable, especially when faced with cumbersome proportions.

Half-Angle Formula

Half-angle formulas are another powerful set of trigonometric identities closely related to double-angle formulas. They are used to express a trigonometric function of half an angle in terms of the function of the original angle. However, in this particular exercise, only the double-angle formula was necessary.
For reference, the half-angle formula for cosine, for instance, is:

• \( \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1+\cos x}{2}} \)

These formulas allow for a similar simplification process as the double-angle formula, just in reverse, for problems where it is useful to reduce rather than expand the angles.
While not needed in our provided problem, mastering when and how to use these formulas enables elegance and efficiency in simplifying various trigonometric expressions, making them essential tools in trigonometry.

Tangent Function

The tangent function, often represented as \(\tan x\), is one of the basic trigonometric functions. It is defined as the ratio of the sine and cosine of an angle \(x\):

• \( \tan x = \frac{\sin x}{\cos x} \)

This function repeats every \(180^\circ\) (or \(\pi\) radians), making it periodic with a period of \(\pi\) radians. The tangent function is notorious for its vertical asymptotes, occurring where \(\cos x = 0\). These asymptotes make the function rise and fall infinitely as it approaches odd multiples of \(90^\circ\) or \(\frac{\pi}{2}\) radians.
In real-world applications, the tangent function can be used to model scenarios involving right triangles, slopes in calculus, and even periodic functions.
In terms of the given exercise, the tangent double-angle formula relied heavily on this fundamental function. Understanding the properties of the tangent function, such as its periodicity and asymptotes, aids significantly in grasping how transformations like those illustrated in the exercise can be achieved, paving the way for further exploration in trigonometry.