Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Fill in the blank(s) to complete each fundamental identity. \(\tan ^{2} x+1=\) ________

Step-by-Step Solution

Answer

\(\sec^2 x\)

Step 1: Identify the relevant trigonometric identity

Recognize that the problem requires us to use one of the Pythagorean identities from trigonometry. The fundamental Pythagorean identity involving tangent is: \( \tan^2 x + 1 = \sec^2 x \). This identity is derived from dividing the primary Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) by \( \cos^2 x \).

Step 2: Simplify the identity if needed

We have identified the correct identity in the previous step. Since it aligns directly with a well-known identity, no additional simplification is necessary.

Step 3: Fill in the blank

Now, apply the identified trigonometric identity. Knowing that \( \tan^2 x + 1 = \sec^2 x \), fill in the blank with \( \sec^2 x \).

Key Concepts

Pythagorean Identities

Within the realm of trigonometry, Pythagorean Identities form a vital component, tying together the trigonometric functions through fundamental equations. These identities stem from the Pythagorean theorem, providing relationships between sine, cosine, and their related functions.

There are three primary Pythagorean identities:

• \(oldsymbol{ ext{sin}^2 x + ext{cos}^2 x = 1} \)
• \(oldsymbol{1 + ext{tan}^2 x = ext{sec}^2 x} \)
• \(oldsymbol{1 + ext{cot}^2 x = ext{csc}^2 x} \)

The identity involving tangent \( ( an^2 x + 1 = ext{sec}^2 x) \) is derived by dividing the basic identity \( ext{sin}^2 x + ext{cos}^2 x = 1\) by \( ext{cos}^2 x\). This approach allows for the transition from sine and cosine to tangent and secant functions.

These identities serve as the baseline in solving trigonometric equations, simplifying complex expressions, and transforming between different trigonometric functions. Understanding these identities deeply enables students to recognize patterns and solve problems more efficiently.

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. Represented by \( ext{tan} \, x \), this function is defined as the ratio of the sine and cosine functions:
\[ ext{tan} \, x = \frac{ ext{sin} \, x }{ ext{cos} \, x} \]
This ratio reveals that the tangent function can become undefined where the cosine function equals zero, leading to its well-known vertical asymptotes.

The tangent function plays a crucial role in various mathematical applications, including:

• Identifying angles and solving right triangles through trigonometric ratios.
• Analyzing periodic phenomena in fields such as physics and engineering.

Its behavior is periodic with a period of \( ext{π} \), allowing it to repeat its pattern every \( 180 \)° or \( ext{π} \text{ radians} \). By understanding the properties of the tangent function and its relationship through Pythagorean identities, students can tackle more advanced trigonometric problems with confidence.

Secant Function

The secant function, denoted as \( ext{sec} \, x \), is the reciprocal of the cosine function. It is represented as:
\[ ext{sec} \, x = \frac{1}{ ext{cos} \, x} \]
This relationship indicates that the secant function will become undefined when the cosine of \( x \) is zero, similar to how the tangent function presents asymptotic behavior in specific intervals.

The secant function is integral in several ways:

• It's involved in deriving trigonometric identities, such as the Pythagorean identity \( ext{tan}^2 x + 1 = ext{sec}^2 x \).
• It aids in understanding the transformations and calculations in real-world applications, like wave motion and alternating current circuits.

Its periodic nature, with a period of \( 2\pi \), helps in periodic analysis, adding depth to its functional aspects and applications.
By appreciating the secant function and its interactions with other trigonometric functions, students gain a comprehensive understanding of trigonometry's scope and practical utility.