Problem 95 - Algebra and Trigonometry with Analytic Geometry

Question

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta\). $$\sqrt{1+\tan ^{2} \theta} ; \quad 3 \pi / 2<\theta<2 \pi$$

Step-by-Step Solution

Answer

\( \sec \theta \) in the interval \( 3\pi/2 < \theta < 2\pi \).

Step 1: Identify the Expression

The given expression is \( \sqrt{1 + \tan^2 \theta} \). We need to rewrite this expression without involving a square root for any value of \( \theta \) in the specified interval.

Step 2: Recognize Trigonometric Identity

Use the Pythagorean identity \( 1 + \tan^2 \theta = \sec^2 \theta \). Therefore, the expression \( \sqrt{1 + \tan^2 \theta} \) can be rewritten as \( \sqrt{\sec^2 \theta} \).

Step 3: Simplify Using Square Root Property

For any real number \( x \), the square root of its square is \( |x| \), i.e., \( \sqrt{x^2} = |x| \). Therefore, \( \sqrt{\sec^2 \theta} = |\sec \theta| \). In this problem, however, we're asked not to use absolute values.

Step 4: Determine the Sign of \( \sec \theta \)

Given \( 3\pi/2 < \theta < 2\pi \), \( \theta \) is in the fourth quadrant where the cosine function is positive. Since \( \sec \theta = 1/\cos \theta \), \( \sec \theta \) will also be positive in this interval.

Step 5: Rewrite Without Absolute Value

Since \( \sec \theta \) is positive in the interval specified, \( |\sec \theta| = \sec \theta \). Thus, the expression \( \sqrt{1 + \tan^2 \theta} \) simplifies to \( \sec \theta \) in the interval \( 3\pi/2 < \theta < 2\pi \).

Key Concepts

Nonradical Expressions

In trigonometry, nonradical expressions require transformation of square root expressions into simpler forms without using radicals. This is essential when simplifying expressions for a particular purpose or context.
Understanding nonradical expressions enables us to eliminate roots in equations and make calculations easier. For example, if you encounter an expression like \( \sqrt{a^2} \), you know it equals \(|a|\), which means we evaluate the absolute value of \(a\).
However, when provided with conditions specifying particular intervals or signs, often the expression can be rewritten without absolute values.

• If \(a\) is positive, \(\sqrt{a^2}\) simplifies to \(a\).
• If negative, it would be \(-a\).
• With clearly specified quadrants for angles, analyzing the positivity or negativity of trigonometric functions is simpler, allowing us to drop absolute values without confusion.

Trigonometric Functions

Trigonometric functions are foundational in the study of angles and periodic phenomena. Used extensively in mathematics, physics, and engineering, these functions help model and solve problems with cyclic patterns.
On the unit circle, each angle corresponds to a point on the circle where trigonometric functions define relationships between the angle's measure and coordinates of the point. Important trigonometric functions include:

• Sine \(\sin\): Ratios comparing opposite side to hypotenuse in a right triangle.
• Cosine \(\cos\): Ratios comparing adjacent side to hypotenuse.
• Tangent \(\tan\): Compares opposite side to adjacent side, \(\tan \theta = \sin \theta / \cos \theta\).
• Secant \(\sec\): The reciprocal of the cosine function, \(\sec \theta = 1 / \cos \theta\).

Understanding these functions is crucial, particularly when evaluating them over different quadrants, which can affect their signs and lead directly to simplified solutions.

Pythagorean Identity

In trigonometry, the Pythagorean identity plays a vital role in simplifying expressions and proving equations. Originating from the Pythagorean Theorem in geometry, these identities relate the squares of the basic trigonometric functions.
The primary Pythagorean identity is \(\sin^2 \theta + \cos^2 \theta = 1\). From this, we can derive that:

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

These identities are not just theoretical; they simplify complex trigonometric expressions. In our original solution, we transformed \(\sqrt{1 + \tan^2 \theta}\) to \(\sec \theta\) by using \(1 + \tan^2 \theta = \sec^2 \theta\). Such transformations allow us to handle calculations easily, and understanding when and how to use these identities is essential for efficient problem-solving in trigonometry.