Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank to complete the trigonometric formula. _______ \(=\frac{1-\cos 2 u}{1+\cos 2 u}\)

Step-by-Step Solution

Answer

The completed formula is \(\tan 2u =\frac{1-\cos 2u}{1+\cos 2u}\).

Step 1: Recognize the formula

One must recognize the formula given in the problem. The formula in this case is a well-known double angle identity, without a name in it.

Step 2: Identify the missing trigonometric identity

To solve this exercise, it needs to be identified which trigonometric identity matches the structure given in the problem. Remembering the identities for trigonometric functions, the missing identity that matches the structure provided is the tangent function. Particularly, this is the identity for the tangent of the double of an angle, given by \(\tan 2u =\frac{1-\cos 2u}{1+\cos 2u}\).

Step 3: Substitute the identified identity into the formula

Lastly, substitute the identified identity, \(\tan 2u\), into the original formula. So, the completed formula becomes \(\tan 2u =\frac{1-\cos 2u}{1+\cos 2u}\).

Key Concepts

Trigonometric Identities

Learning trigonometric identities is essential for students to simplify and solve trigonometry problems effectively. These identities are equations that involve trigonometric functions and are true for any value of the involved angles. They help in transforming one trigonometric form to another, offering a way to make complex calculations more manageable.

A popular subset of these identities is the double angle identities, which express trigonometric functions of double angles in terms of single angles. An example is the identity \(\tan 2u = \frac{1 - \cos 2u}{1 + \cos 2u}\), which connects the tangent of twice an angle to the cosine of the same angle. Double angle identities are instrumental in calculus and are often used to evaluate integrals, solve differential equations, and in many problems involving periodic functions.

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It is the ratio of the sine and cosine functions, or more formally, \(\tan u = \frac{\sin u}{\cos u}\). The tangent represents the slope of the terminal side of an angle traversed in the counterclockwise direction from the positive x-axis on the unit circle.

Understanding the tangent function's behavior is crucial for grasping trigonometry. This function has a period of \(\pi\), which means that it repeats its values every \(\pi\) radians, and it is undefined when the cosine is zero (at odd multiples of \(\frac{\pi}{2}\)). The function exhibits vertical asymptotes, where it reaches infinity, at these undefined points. The tangent function can also be used to solve trigonometric equations and model periodic phenomena, such as sound waves and tides.

Cosine Function

The cosine function, often denoted as \(\cos\), is another core trigonometric function and represents the x-coordinate of a point determined by the angle on the unit circle. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

The cosine function is an even function, which means that \(\cos(-u) = \cos(u)\), and it has a period of \(2\pi\) radians, indicating that it repeats its pattern every \(2\pi\) radians. Knowing the cosine function is crucial when working with oscillations and waves, as it models the regular intervals of crest and trough in a cycle. In practical applications, the cosine function is used in signal processing, engineering, physics and, it plays a pivotal role in describing the motion of pendulums and the generation of alternating current.