Question

$$\text {Sketch the following sets of points.}$$ $$\\{(r, \theta): \pi / 2 \leq \theta \leq 3 \pi / 4\\}$$

Short answer

Based on the provided step by step solution, the short answer is: To sketch the given set of polar coordinates \((r, \theta)\) where \(\pi / 2 \leq \theta \leq 3\pi / 4\), the curve will be in the second quadrant of the coordinate plane. It starts from the vertical axis (y-axis) at \(\theta = \pi/2\) and ends at the line segment \(y=-x\) where \(\theta = 3\pi/4\). The region contains all points corresponding to the polar coordinates given by the range \(\pi / 2 \leq \theta \leq 3\pi / 4\).

Step-by-step solution

Write down the given set of points

The given set of points is: $$\\{(r, \theta): \pi / 2 \leq \theta \leq 3\pi / 4\\}$$

Understand the range of theta

The given range of \(\theta\) is \(\pi / 2 \leq \theta \leq 3\pi / 4\). This range corresponds to the angles in the second quadrant of the coordinate system. Keep in mind that any point in this range can be represented in the form \((r, \theta)\) where \(r\) is the distance from the origin and \(\theta\) is an angle between \(\pi/2\) and \(3\pi/4\).

Convert to Cartesian coordinates

In Cartesian coordinates, any point in polar coordinates \((r, \theta)\) can be represented as \((x, y)\) where: $$x = r\cos\theta$$ $$y = r\sin\theta$$ Since the range of angles in this problem all lie within the second quadrant, the corresponding points in the Cartesian plane will have a positive \(y\) and a negative \(x\) value.

Draw the range of angles

To sketch the range of angles, start by drawing a coordinate plane. In the second quadrant, draw a line segment from the origin to the point where \(\theta = \pi/2\). This point will lie on the vertical (y) axis. Draw another line segment from the origin to the point where \(\theta = 3\pi/4\). This point will lie on the line \(y=-x\). Highlight all the points between these two line segments in the second quadrant, which would represent the polar points for the given range of \(\theta\).

Final sketch

The final sketch should show a region in the second quadrant of the coordinate plane that starts at the vertical axis when \(\theta = \pi/2\) and ends at the line segment \(y = -x\) when \(\theta = 3\pi/4\). This region should contain all points corresponding to the polar coordinates given by the range \(\pi / 2 \leq \theta \leq 3\pi / 4\).

Key concepts

Cartesian Coordinates

Cartesian coordinates are a foundational concept in mathematics, particularly in geometry and trigonometry. This system uses two values, usually termed as \((x, y)\), to specify the location of a point in a two-dimensional space. Here’s how it works:

• The \(x\)-coordinate indicates the horizontal distance of the point from the origin, which is the point where both axes intersect, usually designated as \((0,0)\).
• The \(y\)-coordinate signifies the vertical distance from the origin.

To create a visual representation of any point, you imagine moving horizontally from the origin by \(x\) units and then moving vertically by \(y\) units. This process identifies the exact position of the point in the plane.

Cartesian coordinates are particularly useful for detailed graphing, providing clarity on locations in two-dimensional space. Adjusting from polar to Cartesian coordinates involves using trigonometric functions \(x = r\cos\theta\) and \(y = r\sin\theta\). This is how we translate polar coordinates like \((r, \theta)\) into Cartesian points.

Second Quadrant

The second quadrant is a specific region in the coordinate system. Visualize the coordinate grid divided into four parts, or quadrants. Here's a breakdown of the second quadrant:

• It lies in the upper left part of the plane formed by the \(x\) and \(y\) axes.
• In this region, the \(x\)-values are negative, while the \(y\)-values are positive.

When exploring angles within the second quadrant, you're essentially looking at those between \(\pi/2\) (equal to 90 degrees) and \(\pi\) (or 180 degrees). As mentioned in our exercise, the angle settings of \(\pi/2 \leq \theta \leq 3\pi/4\) confine our points to this quadrant. This means that all the points will have positive \(y\)-coordinates and negative \(x\)-coordinates due to their positioning.

Angle Conversion

Converting angles from one type of measurement to another, such as from degrees to radians or vice versa, is a crucial skill in mathematics. Let's understand how this works:

• Degrees and radians are two methods for measuring angles. While degrees are more common in everyday use, radians are the scientific standard, particularly in higher mathematics.
• One full rotation around a circle is \(360\) degrees, which equals \(2\pi\) radians.

To convert from degrees to radians, you multiply by \(\pi/180\). Conversely, converting from radians to degrees involves multiplying by \(180/\pi\). For example, an angle of \(90\) degrees is equivalent to \(\pi/2\) radians, as can be seen in our problem’s context.

Understanding these conversions helps in visualizing and graphing problems like the one presented, as it allows us to easily situate angles within their appropriate quadrant and coordinate system.