Problem 98 - Algebra and Trigonometry with Analytic Geometry

Question

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\cos ^{2}(\theta / 2)} ; \quad 0<\theta<\pi$$

Step-by-Step Solution

Answer

The expression simplifies to \( \cos(\theta / 2) \).

Step 1: Identify the Expression

We have the expression \( \sqrt{\cos^2(\theta / 2)} \) and we need to rewrite it in nonradical form. The domain given is \( 0 < \theta < \pi \).

Step 2: Simplify the Square Root

The square root of a square expression can be simplified using the principal square root function. Thus, \( \sqrt{\cos^2(\theta / 2)} = |\cos(\theta / 2)| \).

Step 3: Remove Absolute Value Conditions

In the given range \( 0 < \theta < \pi \), the angle \( \theta / 2 \) satisfies \( 0 < \theta / 2 < \pi/2 \). Within this interval, \( \cos(\theta / 2) \) is positive. Therefore, \( |\cos(\theta / 2)| = \cos(\theta / 2) \).

Step 4: Write the Final Expression

Since \( |\cos(\theta / 2)| = \cos(\theta / 2) \) for \( 0 < \theta < \pi \), the expression in nonradical form without absolute values is \( \cos(\theta / 2) \).

Key Concepts

Nonradical Form

When we talk about expressing a mathematical expression in nonradical form, we mean removing any square roots. A radical is essentially a root expression, like square or cube roots. Generally, math problems prefer simpler forms because they are easier to work with in calculations.

To convert an expression from radical to nonradical form, you can use the property that the square root of a square returns the absolute value of the original expression. For example:

• If you have \( \sqrt{x^2} \), it simplifies to \( |x| \).

However, if the expression inside the radical is always non-negative over the desired interval, the absolute value can be omitted. This is an important step in simplifying an expression, ensuring calculations are straightforward and understandable.

Absolute Values

Absolute values measure the distance a number is from zero on a number line, without considering direction. It changes negative values to positive ones, simplifying comparisons and calculations.

When applying absolute values to trigonometric functions like cosine or sine, they play a key role in determining the true value of trigonometric expressions. Normally:

• \(|a| = a, \text{ if } a \geq 0\)
• \(|a| = -a, \text{ if } a < 0\)

While dealing with trigonometric expressions over specific intervals, absolute values simplify understanding of the behavior and values, removing complexities like negative results in otherwise positive contexts.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are invaluable in simplifying expressions and solving equations.

Commonly used identities include:

• Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Double-angle formulas, like \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)

In this particular problem, knowing that \( \sqrt{\cos^2(\theta / 2)} = |\cos(\theta / 2)| \), helps us relate it to the intervals and simplify the form, making the calculations easier. Trigonometric identities bridge the gap in understanding these conversions, making complex trigonometry accessible to solve real-world problems.