Problem 29 - Calculus Volume 3

Question

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\sec t, \quad y=\tan t, \pi \leq t<\frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

The rectangular form is \( x^2 - y^2 = 1 \) with domain \( x < -1 \).

Step 1: Understand the Parametric Equations

We are given the parametric equations: \( x = \sec t \) and \( y = \tan t \). The goal is to eliminate the parameter \( t \) and find a relationship between \( x \) and \( y \) in rectangular form (i.e., involving only \( x \) and \( y \)).

Step 2: Know Pythagorean Identity

Recall the Pythagorean identity for trigonometric functions: \( \sec^2 t = 1 + \tan^2 t \). This identity will help in eliminating the parameter \( t \).

Step 3: Express sec and tan in terms of y

From the given equations, \( y = \tan t \), so \( \tan^2 t = y^2 \). Substitute into the identity: \( \sec^2 t = 1 + y^2 \).

Step 4: Relate x using sec

Since \( x = \sec t \), square both sides to get \( x^2 = \sec^2 t \).

Step 5: Substitute and Formulate Rectangular Equation

Substitute \( x^2 \) for \( \sec^2 t \) in the identity from Step 3: \( x^2 = 1 + y^2 \). Rearrange it to express it in rectangular form: \( x^2 - y^2 = 1 \).

Step 6: Determine Domain of Rectangular Form

The original interval is \( \pi \leq t < \frac{3\pi}{2} \). In this range, \( \sec t \) and \( \tan t \) are both less than zero, which corresponds to the interval where \( x < -1 \). Thus, the domain of the rectangular form is \( x < -1 \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric concept that relates the squares of the sine, cosine, tangent, and secant functions. It is an essential tool when dealing with parametric equations, as it allows us to connect these trigonometric functions with each other and derive relationships between them.

The most common form of the Pythagorean identity is given by:

• \[ \sin^2 t + \cos^2 t = 1 \]
• Another useful form is related specifically to the secant and tangent functions:\[ \sec^2 t = 1 + \tan^2 t \]

In the context of transforming parametric equations to rectangular form, this identity can help eliminate the parameter \( t \). By knowing that \( x = \sec t \) and \( y = \tan t \), you substitute these values into the Pythagorean identity for secant and tangent to create a new equation involving only \( x \) and \( y \). This equation is the key to unlocking the rectangular form.

Trigonometric Functions

Trigonometric functions like sine, cosine, tangent, and secant are core components of trigonometry. They describe the relationships between the angles and sides of triangles and are foundational in converting parametric to rectangular equations.

Here are some key points about these functions:

• **Sine and Cosine**: These functions are the ratios of the length of sides in a right triangle. They are defined as \( \sin t = \frac{\text{opposite}}{\text{hypotenuse}} \) and \( \cos t = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• **Tangent and Secant**: Tangent is the ratio \( \tan t = \frac{\sin t}{\cos t} \), and secant is the reciprocal of cosine, \( \sec t = \frac{1}{\cos t} \).
• These functions have specific ranges and behaviors based on the angle \( t \), which can help determine the domain when converted into a rectangular form.

Understanding these functions helps in visualizing and manipulating trigonometric identities, as seen in this problem where we relate \( \sec t \) and \( \tan t \) using the Pythagorean identity to derive a relationship between \( x \) and \( y \).

Domain of a Function

The domain of a function defines the set of allowable input values (or \( x \)-values) that a function can accept. When dealing with parametric equations, converting to rectangular form often results in a new expression with its own domain.

In the exercise, the provided parameter \( t \) is restricted to the interval \( \pi \leq t < \frac{3\pi}{2} \). Within this interval:

• The secant function \( \sec t \) is always negative because \( t \) is in the third quadrant where cosine is negative and secant (being the reciprocal of cosine) follows the same sign. Thus, \( x = \sec t < -1 \).
• The tangent function \( \tan t \) is also negative because sine and cosine are both negative in the third quadrant, resulting in a positive quotient \( \tan t = \frac{\sin t}{\cos t} \).

Thus, the domain of the rectangular form \( x^2 - y^2 = 1 \) strictly needs to account for these conditions. As a result, the domain of this rectangular form will constrain \( x \) such that \( x < -1 \), reflecting the restricted range of \( t \) given in the original parametric equations.