Problem 75 - Precalculus with Limits

Question

In Exercises 73-76, use the half-angle formulas to simplify the expression. \( \sqrt{\dfrac{1 - \cos 8x}{1 + \cos 8x}} \)

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt{\dfrac{1 - \cos 8x}{1 + \cos 8x}} \) is \( \tan{4x} \).

Step 1: Identify the half-angle formula that fits

From the half-angle formula list, we identify that the required formula to use is \[ \tan{x} = \sqrt{\dfrac{1 - \cos{2x}}{1 + \cos{2x}}} \]. In this formula, replace \( 2x \) with \( 8x \) to match with the given expression.

Step 2: Apply the formula to the expression

Now replace the given expression with the identified formula. So, \( \sqrt{\dfrac{1 - \cos 8x}{1 + \cos 8x}} \) can be written as \( \tan{4x} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships involving trigonometric functions like sine, cosine, and tangent. These identities help simplify complex trigonometric expressions and solve equations more easily. Half-angle identities, a specific type of trigonometric identity, express trigonometric functions of half an angle in terms of the trigonometric functions of the original angle.

For example, the half-angle formula for tangent is:

• \[ \tan{x} = \sqrt{\dfrac{1 - \cos{2x}}{1 + \cos{2x}}} \]

In this expression, the tangent of angle \( x \) is represented in terms of cosine of double that angle, \( 2x \). This formula is particularly useful when dealing with equations involving \( \tan{x} \), especially when simplifying expressions or calculating values without direct angle measurements. By re-writing expressions with these identities, we can often uncover solutions that aren't immediately obvious.

Trigonometry Simplification

Simplifying trigonometric expressions is a crucial skill that allows one to solve problems more efficiently. Using identities, like the half-angle identities, helps reduce complicated expressions into simpler forms. When simplifying, our main goal is to express a trigonometric problem in a form that's easier to work with or solve.

In the exercise given, the expression \( \sqrt{\dfrac{1 - \cos{8x}}{1 + \cos{8x}}} \) might look complex at first glance. However, by recognizing and applying the half-angle formula for tangent, this expression is simplified to \( \tan{4x} \). This dramatically reduces the problem's complexity by condensing it into a familiar trigonometric function. Simplifying trigonometric expressions helps in various applications, from solving equations to handle calculus problems, and is an essential strategy in any math student's toolbox.

Cosine Functions

Cosine is one of the primary trigonometric functions, closely related to sine and tangent. It describes the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

• The cosine function is periodic, meaning it repeats its values in regular intervals, specifically every \( 2\pi \) for the functions like \( \cos{x} \).
• In trigonometric identities and formulas, it's crucial to note that even slight changes in the argument of cosine, like \( 8x \) instead of \( x \), can significantly alter the expression.

Through identities, the cosine function can be manipulated to reveal new relationships, such as in our exercise where \( \cos{8x} \) transforms under the square root into an expression for \( \tan{4x} \). Understanding cosine deeply not only helps in solving trigonometric equations but also provides foundational knowledge for calculus and analytical geometry.