Problem 97 - Algebra and Trigonometry with Analytic Geometry

Question

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\sin ^{2}(\theta / 2)}, \quad 2 \pi<\theta<4 \pi$$

Step-by-Step Solution

Answer

The nonradical form is \(-\sin(\theta/2)\).

Step 1: Simplify the Expression

We start by simplifying the expression \(\sqrt{\sin ^{2}(\theta / 2)}\). This is equivalent to the absolute value of \(\sin(\theta/2)\), which can be written as \(|\sin(\theta/2)|\).

Step 2: Determine the Sign of the Sine Function

We need to find whether \(\sin(\theta/2)\) is positive or negative in the given range \(2\pi < \theta < 4\pi\). Since \(\theta/2\) is in the range \(\pi < \theta/2 < 2\pi\), \(\sin(\theta/2)\) is negative in this range.

Step 3: Express the Absolute Value

Since \(\sin(\theta/2)\) is negative, the absolute value can be written as \(-\sin(\theta/2)\).

Step 4: Write the Final Expression

Thus, the nonradical form of the given expression is \(-\sin(\theta/2)\).

Key Concepts

Nonradical Forms

When we talk about rewriting an expression in its nonradical form, we are essentially looking to represent it without square roots or radicals. In many cases, this involves using absolute values. For instance, the expression \(\sqrt{\sin^2(\theta/2)}\) can be rewritten using absolute values as simply \(|\sin(\theta/2)|\). This means we express the magnitude of the sine function without considering its direction, positive or negative.
Nonradical forms are useful for simplifying and solving equations, especially in calculus and trigonometry. By converting expressions into a nonradical form, mathematicians often find it easier to handle complex trigonometric functions.
Understanding nonradical expressions helps in visualizing how the function behaves across different ranges of angles. It's crucial when determining solutions in trigonometric identities, making calculations clearer and more straightforward.

Absolute Value Properties

Understanding absolute value properties is essential for grasping the concept of rewriting expressions like \(|\sin(\theta/2)|\). Absolute value, denoted by vertical bars \(|...|\), represents the distance of a number from zero on the number line.
Here are some basic properties:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.
• Absolute values are always non-negative.

In trigonometry, when dealing with expressions like \(|\sin(\theta/2)|\), absolute value helps clarify whether the sine function results in positive or negative values over particular intervals of \(\theta\). In our exercise, since \(\sin(\theta/2)\) is negative within a specific range, we simplify it as \(-\sin(\theta/2)\) to rid the expression of the absolute value, thanks to these properties.

Sine Function

The sine function, one of the fundamental trigonometric functions, describes the y-coordinate of a point on the unit circle corresponding to a given angle. When we represent an angle \(\theta\) in the coordinate plane, \(\sin(\theta)\) shows how far up or down the point lies from the x-axis.
Key points about the sine function include:

• It is periodic with a period of \(2\pi\).
• The function oscillates between -1 and 1.
• It is positive in the first and second quadrants and negative in the third and fourth.

In the given exercise, understanding the sine function in terms of its positive and negative values was critical. Over the interval where \(2\pi < \theta < 4\pi\), it creates a scenario where \(\theta/2\) is between \(\pi\) and \(2\pi\), causing \(\sin(\theta/2)\) to be negative. This understanding allows us to use the expression \(-\sin(\theta/2)\) in solving the problem.

Angle Transformations

Angle transformations refer to the modifications made to angles for simplifying trigonometric expressions or solving equations. These transformations can involve division, multiplication, or translation actions that convert a given angle to a more manageable form.
In our exercise, we dealt with the transformation \(\theta/2\). This simply means the original angle \(\theta\) is halved, impacting the range and function outputs we're dealing with. The transformed angle \(\theta/2\) falls within an interesting range, meaning it represents angles from \(\pi\) to \(2\pi\) in the trigonometric circle.
These transformations are essential:

• They help isolate important intervals where functions change nature, as seen with the sine function becoming negative.
• They simplify the process of evaluating trigonometric expressions, aiming to achieve simpler forms or solutions.

A solid grasp on angle transformations enables students to effectively engage with more complex trigonometric identities and expressions, like the one in the exercise provided, without hesitation.