Problem 45 - Contemporary Precalculus

Question

Find a rectangular equation that is equivalent to the given polar equation. \(r=\sec \theta[\text {Hint}:\) Express the right side in terms of cosine.]

Step-by-Step Solution

Answer

Question: Convert the polar equation \(r=\sec\theta\) to a rectangular equation. Answer: The rectangular equation equivalent to the given polar equation is \(x=1\).

Step 1: Expressing the right side in terms of cosine

Recall that \(\sec\theta = \frac{1}{\cos\theta}\). So, we have \(r = \frac{1}{\cos\theta}\).

Step 2: Multiply by the denominator

Multiply both sides of the equation by \(\cos\theta\) to get: \(r\cos\theta = 1\)

Step 3: Convert to rectangular coordinates

Now, replace \(r\cos\theta\) with \(x\) in the equation above: \(x = 1\) This is a rectangular equation equivalent to the given polar equation: \(r=\sec\theta\).

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin in Cartesian coordinates) is called the pole, and the ray from the pole in the reference direction is the polar axis.

The polar coordinates are written as a pair \(r, \theta\), where \(r\) is the radius or the distance from the pole, and \(\theta\) is the angle measured in radians from the polar axis. Converting between polar and rectangular coordinates involves trigonometric functions, as seen in converting \(r=\frac{1}{\(cos\theta\)}\) to a rectangular equation.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are another way of representing points in a plane using two perpendicular lines; one horizontal (x-axis) and one vertical (y-axis). Each axis has a scale, and the position of any point is given by two numbers, commonly referred to as \(x\) and \(y\).

To convert from polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \) you use the formulas \( x = r\cos\theta \) and \( y = r\sin\theta \). When solving the equation from the exercise, the trigonometric identity \( r\cos\theta = x \) was used to transform the polar equation \(r=\frac{1}{\cos\theta}\) into the rectangular form \(x = 1\).

Trigonometric Functions

Trigonometric functions are fundamental in the study of periodic phenomena, including waves, oscillations, and circles. They relate the angles of a triangle to the lengths of its sides. The basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), and their reciprocals: cosecant (csc), secant (sec), and cotangent (cot).

For example, \(\cos\theta\) is the ratio of the adjacent side to the hypotenuse in a right-angled triangle, and \(\sec\theta\), which is used in the exercise, is the reciprocal of \(\cos\theta\) hence \(\sec\theta = \frac{1}{\cos\theta}\). Understanding these functions is vital for converting between polar and rectangular equations.

Precalculus

Precalculus is a course that prepares students for calculus, and it includes topics such as functions, complex numbers, trigonometry, vectors, and coordinate systems. It introduces the foundational concepts needed to understand the more advanced mathematics of calculus.

In the context of the given exercise, precalculus provides the tools necessary to understand how polar and rectangular coordinates are related and how to work with trigonometric functions to manipulate and convert between different forms of equations.