Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

x=4 \sin t, y=4 \cos t-5 ; 0 \leq t \leq \pi

Step-by-Step Solution

Answer

The equation describes the top half of a circle centered at (0, -5) with a radius of 4.

Step 1: Understand the Parametric Equations

The given equations are parametric equations for a set of points. They express the coordinates \((x, y)\) in terms of a parameter \(t\). Specifically, \(x = 4 \sin t\) and \(y = 4 \cos t - 5\) for \(0 \leq t \leq \pi\).

Step 2: Identify the Range of t

The parameter \(t\) varies from 0 to \(\pi\). This represents the upper half of a circle since the sine and cosine functions describe circular motion when plotted parametrically.

Step 3: Substitute and Simplify

Using the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\), we substitute to find \(x^2 + (y+5)^2\). Replace: \(\sin t = \frac{x}{4}\) and \(\cos t = \frac{y+5}{4}\). Therefore, \(x^2 = 16\sin^2 t\) and \((y+5)^2 = 16\cos^2 t\).

Step 4: Derive the Relationship Between x and y

Use the identity from Step 3: \((\frac{x}{4})^2 + (\frac{y+5}{4})^2 = 1\). Multiply by 16 to simplify: \(x^2 + (y+5)^2 = 16\). This describes a circle centered at \((0, -5)\) with radius 4.

Step 5: Identify the Specific Area of the Circle

Since \(t\) ranges from 0 to \(\pi\), only the top half of the circle is plotted, which represents a semicircle above the line \(y = -5\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential mathematical tools that relate the angles and sides of triangles to each other through trigonometric functions like sine, cosine, and tangent. One of the most fundamental identities, which we use in this exercise, is the Pythagorean identity:

• This identity states that for any angle \(t\), \(\sin^2 t + \cos^2 t = 1\).
• In our exercise, this identity helps us connect the parametric equations \(x = 4 \sin t\) and \(y = 4 \cos t - 5\) to form a circle equation.
• By squaring these equations and substituting them into the Pythagorean identity, we confirm that \((\frac{x}{4})^2 + (\frac{y+5}{4})^2 = 1\).

These relationships are pivotal for transitioning from parametric to Cartesian equations, revealing more about the graph's structure.

Circular Motion

Circular motion in mathematics often involves tracing the path of a point in a circular trajectory. In parametric equations like the ones given, \(x = 4 \sin t\) and \(y = 4 \cos t - 5\), they describe how a point moves along a circular path as the parameter \(t\) changes.

• The sine and cosine functions are periodic, meaning they repeat their values in regular intervals, which is characteristic of circular motion.
• As \(t\) varies from 0 to \(\pi\), the equations describe the upper half of a circle, hence tracing a semicircle path.

This reveals the role of trigonometric functions in modeling repetitive and balanced motion, such as the oscillation of a pendulum or the rotation of a wheel.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometric relationships and greatly assists in linking parametric forms to geometric figures. The identity \(\sin^2 t + \cos^2 t = 1\) is derived from the geometric interpretation of the unit circle:

• The unit circle is a circle centered at the origin with a radius of one. It helps visualize how sine and cosine depend on angles \(t\) by creating a right triangle within the circle.
• In our specific example, these identities are used by expressing sine and cosine in terms of the parameters \(x\) and \(y\), leading to \(x^2 = 16\sin^2 t\) and \((y+5)^2 = 16\cos^2 t\).

This identity, thus, ensures consistency and accuracy in converting between trigonometric forms and Cartesian coordinates.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to describe geometric shapes using algebra and the coordinate plane. In this exercise, converting parametric equations \(x = 4 \sin t\) and \(y = 4 \cos t - 5\) into Cartesian coordinates reveals:

• A circle equation \(x^2 + (y+5)^2 = 16\), which can be graphically represented in the coordinate plane.
• The circle is centered at \((0, -5)\) with a radius of 4, depicting how geometric properties are deduced from algebraic manipulations.
• Considering the interval of \(t\), we note that only the semicircle above the line \(y = -5\) is covered, as \(t\) ranges from 0 to \(\pi\).

Therefore, coordinate geometry not only visualizes algebraic expressions but also aids in comprehending and predicting the shapes and areas these equations represent.