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Chapter 6: Problem 91

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=4 \sin \theta$$

Short Answer

Expert verified
The rectangular form of the given polar equation is \(x^2 + y^2 = 4y\).

Step by step solution

01

Express r in rectangular form

Starting with the given polar equation \(r = 4 \sin \theta\), recall that in rectangular coordinates, \(r = \sqrt{x^2 + y^2}\). Thus, we can write the equation as \(\sqrt{x^2 + y^2} = 4 \sin \theta\).
02

Express sine theta in rectangular form

Now we know that in rectangular coordinates, \(\sin \theta = y / r\). Substituting this into the equation obtained in step 1 gives us \(\sqrt{x^2 + y^2} = 4 \cdot \frac{y}{\sqrt{x^2 + y^2}}\).
03

Simplify the equation

To simplify the equation further, multiply each side by \(\sqrt{x^2 + y^2}\) to eliminate the square root on the right side. We thus obtain the equation \(x^2 + y^2 = 4y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a way of representing points in a plane using a combination of a radius and an angle. This system is different from the more familiar rectangular (or Cartesian) coordinates, which use an x and y value to represent a point.
Think of polar coordinates as drawing a circle. You start at the center (origin) and use a line to reach out to the edge of the circle. The length of this line is the radius ( "), and the angle it makes with the positive x-axis is \(\theta\).
  • The radius, \(r\), tells you how far the point is from the origin.
  • The angle, \(\theta\), tells you the direction from the positive x-axis.
For example, if you have polar coordinates \( (3, \frac{\pi}{4})\), it means you move 3 units from the origin in the direction \frac{\pi}{4} radians from the positive x-axis.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two distinct values: x and y. These values correlate directly with the point鈥檚 horizontal and vertical positions, respectively.
Think of it as drawing on graph paper. The \(x\) value tells you how far to move left or right from the origin, while the \(y\) value lets you know how much to move up or down.
  • The x-coordinate represents horizontal distance from the y-axis.
  • The y-coordinate represents vertical distance from the x-axis.
In this system, a point located at \( (3, 4) \) means moving 3 units to the right and 4 units up from the origin on the graph.
Trigonometric Identities
Trigonometric identities are equations that relate the angles and sides of triangles to the functions sine, cosine, and tangent. These identities can be used to simplify complex equations, especially when you need to convert from polar to rectangular coordinates.
A key identity at play here is the relationship involving \(\sin(\theta)\). In the context of conversion, \(\sin(\theta)\) is equivalent to \(\frac{y}{r}\) in rectangular terms.
  • This helps convert polar equations like \(r = 4\sin(\theta)\) to their rectangular forms, as you substitute \(y/r\) in place of \(\sin(\theta)\).
  • Multiplying through by \(\sqrt{x^2 + y^2}\) can help clear radicals and easily transition to a rectangular format.
Understanding these identities and relationships is crucial in mathematics as they bridge the gap between polar and rectangular views, aiding in visualization and solving various equations.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. To find the angle between two lines whose angles of inclination \(\theta_{1}\) and \(\theta_{2}\) are known, substitute \(\theta_{1}\) and \(\theta_{2}\) for \(m_{1}\) and \(m_{2},\) respectively, in the formula for the angle between two lines.

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$x^{2}+y^{2}=a^{2}$$

True or False? Determine whether the statement is true or false. Justify your answer. The conic represented by the following equation is a parabola. \(r=\frac{6}{3-2 \cos \theta}\)

Write a short paragraph explaining why parametric equations are useful.

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$3 x-y+2=0$$
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