Problem 54 - Precalculus: Functions and Graphs

Question

Show that the equation is not an Identity. $$\sec t=\sqrt{\tan ^{2} t+1}$$

Step-by-Step Solution

Answer

The equation is not an identity as it fails for certain values, e.g., \( t = \frac{\pi}{2} \).

Step 1: Recall the Definitions

Start by recalling the definitions of the functions involved. The secant function is defined as \( \sec t = \frac{1}{\cos t} \). The tangent function is defined as \( \tan t = \frac{\sin t}{\cos t} \).

Step 2: Simplify the Right Side

Simplify the expression \( \sqrt{\tan^2 t + 1} \). Using the definition, \( \tan^2 t = (\frac{\sin t}{\cos t})^2 \). So, \( \tan^2 t + 1 = \frac{\sin^2 t}{\cos^2 t} + 1 \).

Step 3: Combine Terms Under a Common Denominator

Combine \( \frac{\sin^2 t}{\cos^2 t} + 1 \) under a single denominator: \( \frac{\sin^2 t + \cos^2 t}{\cos^2 t} \).

Step 4: Apply Pythagorean Identity

Recall that \( \sin^2 t + \cos^2 t = 1 \), so we substitute this into our expression to get \( \frac{1}{\cos^2 t} \).

Step 5: Simplify the Full Expression

Simplifying \( \sqrt{\frac{1}{\cos^2 t}} \) yields \( \frac{1}{\cos t} \), which is equal to \( \sec t \).

Step 6: Compare Both Sides

Both the left side and right side simplify to \( \sec t \), so at a cursory glance this seems like an identity.

Step 7: Test Specific Values

However, let's check specific values. For example, at \( t = 0 \), both sides are equal (1 = 1). Now, consider \( t = \frac{\pi}{2} \). Here, \( \sec(\frac{\pi}{2}) = 1 \), whereas \( \sqrt{\tan^2(\frac{\pi}{2}) + 1} \) is undefined, since \( \tan(\frac{\pi}{2}) \) is not defined.

Step 8: Conclude the Identity Test

Since the equation does not hold for \( t = \frac{\pi}{2} \), this demonstrates that the given equation is not an identity.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec t \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function:

• \( \sec t = \frac{1}{\cos t} \)

This means that wherever cosine is zero, the secant function becomes undefined. For instance, at angles like \( t = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \), the cosine function equals zero, leading to points where the secant function cannot be calculated.

The secant function is particularly important in trigonometry because it can simplify complex equations and is widely used in various applications such as calculus, physics, and engineering. It's crucial to understand its behavior, especially where it becomes undefined, which often leads to regions of interest or discontinuities in functions.

Tangent Function

The tangent function, denoted as \( \tan t \), is the ratio of the opposite side to the adjacent side in a right triangle. In terms of trigonometric identities, tangent is expressed as:

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

This ratio, like the secant, can also become undefined. For example, when \( \cos t = 0 \), \( \tan t \) is undefined, which typically occurs at odd multiples of \( \frac{\pi}{2} \).

Understanding the tangent function is essential because it helps in transforming expressions into different forms for easier analysis or solving. It is also frequently used in derivative and integration processes in calculus.

Pythagorean Identity

The Pythagorean identity is a fundamental relation between the sine and cosine functions. It states that for any angle \( t \):

• \( \sin^2 t + \cos^2 t = 1 \)

This identity is derived from the Pythagorean theorem and is central to many trigonometric simplifications and proofs.

In solving trigonometric equations, particularly the one given in the original exercise, the Pythagorean identity plays a key role. By recognizing that \( \sin^2 t + \cos^2 t = 1 \), the problem simplifies greatly, revealing equivalences or contradictions in expressions.
The identity highlights the inherent relationship and constraints shared by sine and cosine, and also establishes the fundamental properties of trigonometry on which many other identities are based.

Trigonometric Functions

Trigonometric functions, including sine, cosine, tangent, cosecant, secant, and cotangent, are the building blocks of trigonometry. They define the relationships between the angles and sides of triangles.

• Sine (\( \sin \)) and cosine (\( \cos \)) are the ratios of sides in a right-angled triangle.
• Secant (\( \sec \)), tangent (\( \tan \)), and other functions are derived from these primary functions.

These functions extend beyond the scope of triangles and are used to model periodic phenomena such as waves and oscillations.

Having a robust understanding of trigonometric functions is crucial for solving complex mathematical problems. Their applications are vast, appearing in physics for wave behavior analyses, in engineering for circuit design, and even in computer graphics for rendering images.
Trigonometric functions provide the framework for connecting algebraic expressions with geometric interpretations, making them indispensable tools in advanced mathematics.