Problem 93 - Algebra and Trigonometry with Analytic Geometry

Question

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{\sec ^{2} \theta-1} ; \quad \pi / 2<\theta<\pi$$

Step-by-Step Solution

Answer

The expression becomes \(-\tan \theta\).

Step 1: Identify the Trigonometric Identity

Recognize that the expression \(\sec^2 \theta - 1\) can be rewritten using the trigonometric identity \(\sec^2 \theta = 1 + \tan^2 \theta\). Therefore, \(\sec^2 \theta - 1 = \tan^2 \theta\).

Step 2: Rewrite the Expression

Substitute \(\tan^2 \theta\) for \(\sec^2 \theta - 1\) in the square root. The expression becomes \(\sqrt{\tan^2 \theta}\).

Step 3: Simplify the Square Root

Recognize that \(\sqrt{\tan^2 \theta} = |\tan \theta|\). However, because the condition given is \(\pi/2 < \theta < \pi\), and in this interval, \(\tan \theta\) is negative, we have \(|\tan \theta| = -\tan \theta\).

Step 4: Final Expression

Using the above analysis, rewrite the expression as \(-\tan \theta\) without using absolute values.

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions. Like its fellow trigonometric functions, it originates from the basic circle geometry. The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. This means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding this relationship is crucial because it allows us to rewrite expressions involving the secant function in terms of sine and cosine, which are often more familiar.
In the problem at hand, we encounter \( \sec^2 \theta \). Utilizing trigonometric identities, we can express \( \sec^2 \theta \) using the tangent function:

• \( \sec^2 \theta = 1 + \tan^2 \theta \)

This identity helps simplify expressions like \( \sec^2 \theta - 1 \), leading directly to the tangent function.

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is another essential part of trigonometry. It can be defined in several equivalent ways, most commonly as the ratio of the sine and cosine functions:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This definition can be incredibly helpful in many mathematical scenarios, especially when manipulating expressions and solving equations.
For angles in the interval \( \pi/2 < \theta < \pi \), it is important to note that the cosine is negative, leading to a negative tangent value.

• Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cos \theta < 0 \), hence, \( \tan \theta < 0 \).

Recognizing the behavior of \( \tan \theta \) in different quadrants allows one to correctly simplify expressions, as shown in our example where \( \sqrt{\tan^2 \theta} = -\tan \theta \) because \( \tan \theta \) is negative in the specified interval.

Simplifying Expressions

Simplifying expressions in trigonometry often relies on recognizing key identities and understanding the behavior of trigonometric functions in various intervals. In our exercise, the expression \( \sqrt{\sec^2 \theta - 1} \) is simplified using basic trigonometric identities.
Here’s the breakdown of the simplification:

• Recognize the identity \( \sec^2 \theta - 1 = \tan^2 \theta \).
• Substitute into the expression: \( \sqrt{\tan^2 \theta} \).
• Utilize the interval \( \pi/2 < \theta < \pi \), knowing that \( \tan \theta \) is negative in this range.
• Conclude that \( \sqrt{\tan^2 \theta} = -\tan \theta \) to avoid using absolute values.

This simplification process involves transforming a sometimes unfamiliar expression into a recognizable and easily computable form.