Problem 68 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(y=x\)

Step-by-Step Solution

Answer

The polar form of rectangular equation \(y=x\) is \(\theta = 45°\) or \(\theta = 45° + k*180°\), where k is an integer.

Step 1: Express x and y in terms of r and θ

To convert the equation \(y=x\) from rectangular to polar form, start by replacing \(x\) with \(r\cos(\theta)\) and \(y\) with \(r\sin(\theta)\). So the equation becomes \(r\sin(\theta) = r\cos(\theta)\)

Step 2: Simplify the equation

To simplify this equation, observe that we can divide both sides of the equation by \(r\), provided that \(r≠0\) (which is a fair assumption, because for \(r=0\), \(x=y=0\) which is a point that fulfills the original \(y=x\) anyway). This division yields \(\sin(\theta)=\cos(\theta)\)

Step 3: Convert cos to sin

We know from trigonometry that \(\cos(\theta) = \sin(90°-\theta)\). Using this relation, the equation becomes \(\sin(\theta) = \sin(90°-\theta)\)

Step 4: Finalize conversion

The relationship between θ from two angles is that they either equal each other or sum to 180°. Therefore, two possible solutions are \(\theta = 45°\) or \(\theta = 45° + k*180°\), where k is an integer value

Key Concepts

Rectangular to Polar Conversion

Converting between rectangular and polar coordinates is crucial for understanding and working with different systems of equations and graphs. In rectangular coordinates, a point is represented by the coordinates \((x, y)\), whereas in polar coordinates, the same point is described by the distance from the origin \(r\) and the angle \(\theta\) from the positive x-axis.

For conversion, one can use the relations:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

In our original exercise, the equation \(y = x\) is represented in rectangular form. To convert it to polar form, replace \(x\) and \(y\) with \(r\cos(\theta)\) and \(r\sin(\theta)\) respectively. This gives us the equation \(r\sin(\theta) = r\cos(\theta)\).

By simplifying, we divide both sides by \(r\), resulting in \(\sin(\theta) = \cos(\theta)\), which moves the problem into a more manageable trigonometric space.

Trigonometric Identities

Trigonometric identities are fundamental tools for simplifying expressions, solving equations, and proving relationships in trigonometry. They relate various trigonometric functions, allowing transformation and manipulation of equations.

In the solution to the equation \(\sin(\theta) = \cos(\theta)\), we can use the identity \(\cos(\theta) = \sin(90^\circ - \theta)\). This is a result of the complementary nature of sine and cosine, highlighting that the sine of an angle is the cosine of its complement.

Applying this identity transforms the equation into \(\sin(\theta) = \sin(90^\circ - \theta)\), indicating that there are two possible solutions: either the angles are equal or they sum to \(180^\circ\).

• This results in the polar angle solutions \(\theta = 45^\circ\)
• Or \(\theta = 45^\circ + k \times 180^\circ\), where \(k\) is any integer

These solutions showcase how trigonometric identities can expand the range of possible results by considering periodic and complementary properties of angles.

Polar Equations

Polar equations are expressions derived in polar coordinates, with \(r\) (the radial distance) and \(\theta\) (the angle) as variables. In this exercise, once the rectangular equation \(y = x\) is converted to polar coordinates, it becomes \(\sin(\theta) = \cos(\theta)\), a relationship solely dependent on angle \(\theta\).

Unlike rectangular equations that describe relationships in linear terms, polar equations often describe curves or shapes naturally, like circles or spirals. The location of points relies on the angle and distance from the origin, providing a different yet equally powerful means of geometrical representation.

In this case, the polar equation \(\theta = 45^\circ\) describes a line radiating from the origin at a \(45^\circ\) angle. It showcases the power of polar representation to illustrate direction and angle, crucial for fields like physics and engineering where direction and magnitude are essential.

Understanding and practicing conversions between rectangular and polar forms allows for versatility in approaching mathematical problems and recognizing scenarios where one may be more advantageous to use than the other.