Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\cos \frac{x}{2}, \text { given } \cos x=\frac{1}{4} \text { and } 0 < x < \frac{\pi}{2}$$

Step-by-Step Solution

Answer

\( \cos\left(\frac{x}{2}\right) = \frac{\sqrt{10}}{4} \).

Step 1: Identify the half-angle identity for cosine

The half-angle identity for cosine is given by the formula \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \). This identity will help find the expression for \( \cos\frac{x}{2} \).

Step 2: Determine the appropriate sign

Since \( x \) is between \( 0 \) and \( \frac{\pi}{2} \), \( \frac{x}{2} \) is between \( 0 \) and \( \frac{\pi}{4} \). In this interval, \( \cos \left( \frac{x}{2} \right) \) is positive.

Step 3: Substitute \( \cos x \) into the half-angle identity

Substitute \( \cos x = \frac{1}{4} \) into the half-angle identity: \( \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \frac{1}{4}}{2}} \).

Step 4: Simplify the expression

Simplify \( \sqrt{\frac{1 + \frac{1}{4}}{2}} \) to find \( \cos\left(\frac{x}{2}\right) \).\[\cos\left(\frac{x}{2}\right) = \sqrt{\frac{\frac{5}{4}}{2}} = \sqrt{\frac{5}{8}} = \frac{\sqrt{5}}{2\sqrt{2}} \]Further simplify by multiplying numerator and denominator by \( \sqrt{2} \) to rationalize the denominator:\[\frac{\sqrt{5}}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{4}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables where both sides of the equation are defined. These identities are foundational in solving trigonometric problems and proving relationships. One of the key identities related to our problem is the half-angle identity for cosine.

The general half-angle identity for cosine is \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \). This identity allows us to find the cosine of half an angle if we know the cosine of the angle itself. It's called a "half-angle" identity because it simplifies and splits the angle \(x\) in half, making it easier to handle.

In every problem, it's crucial to choose the correct sign, positive or negative, which depends on the range of the angle. This ensures the solution respects the values in the unit circle for the specific angle interval you are working with.

Cosine Function

The cosine function, one of the primary trigonometric functions, is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It can be written as \( \cos(\theta) \) where \( \theta \) is the angle. Cosine values vary between -1 and 1, and the function is periodic with a period of \(2\pi\).

The cosine function has several useful identities, like the half-angle identity used here, to simplify the process of calculating cosines at specific angles. In our problem, where we need to determine \( \cos\left(\frac{x}{2}\right) \), we start with knowing \( \cos x = \frac{1}{4} \). This small piece of information is powerful as it anchors our calculations to this known value, allowing us to develop further insights using familiar identities.

The relevance of \( \cos(\theta) \) in trigonometry reaches beyond triangles. It extends into wave functions, oscillations, and even into quantum mechanics, illustrating its versatility and pivotal role in mathematics.

Angle Simplification

Angle simplification is a technique used in trigonometry to reduce complex angles into simpler, more manageable forms. This often involves using identities that relate an angle to its half or double, like the half-angle identity we've used.

To simplify angles means to express them in a form that's easier to calculate or more convenient for further analysis. In the problem presented, we were given \( x \) within a certain interval and needed to find \( \cos\left(\frac{x}{2}\right) \). By utilizing the half-angle identity, the initial expression is transformed and simplified to a solvable form. This simplification is crucial in reducing computational complexity.

While calculating \( \cos x \) was straightforward based on the provided value, transforming it to \( \cos\left(\frac{x}{2}\right) \) required the application of algebraic steps like rationalizing the denominator, leading to the final simplified value. Thus, angle simplification isn't just about reducing the size of an angle; it's a gateway to discovering more about that angle's trigonometric properties.