Problem 17 - Algebra 2 and Trigonometry

Question

The coordinates of a point are given. a. Find the distance of the point from the origin. Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains the given point. \((0,7)\)

Step-by-Step Solution

Answer

Distance is 7; angle is 90°.

Step 1: Identify the Coordinates

The given point is \((0, 7)\), which means the x-coordinate is 0 and the y-coordinate is 7.

Step 2: Calculate the Distance from the Origin

The distance \(d\) from the point to the origin can be found using the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here, the origin is \((0,0)\), so:\[d = \sqrt{(0 - 0)^2 + (7 - 0)^2} = \sqrt{0 + 49} = \sqrt{49} = 7\].

Step 3: Calculate the Angle in Standard Position

To find the angle \(\theta\) in standard position, we use the tangent function:\[\tan(\theta) = \frac{y}{x}\].Since \(x = 0\), the line is vertical, and the angle with the x-axis (positive direction) is \(90^\circ\).

Step 4: Summarize Results

The distance of the point \((0,7)\) from the origin is 7 units, and the angle in standard position is \(90^\circ\).

Key Concepts

Distance Formula

The Distance Formula is a fundamental concept in coordinate geometry that helps determine the length between two points on a plane. It's rooted in the Pythagorean theorem and is particularly useful when you have the coordinates of two points.

The general form of the Distance Formula is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(d\) is the distance between these two points.

In our exercise, the coordinates of the point are \((0,7)\), and the origin is \((0,0)\). Plugging these into the formula, we find:\[d = \sqrt{(0 - 0)^2 + (7 - 0)^2} = \sqrt{49} = 7\]

This tells us that the distance from the point to the origin is 7 units, which shows the practical application of the Distance Formula.

Angle in Standard Position

An Angle in Standard Position is considered to be in standard position when its vertex is at the origin (\(0,0\)) of the coordinate plane, and one side (the initial side) lies along the positive x-axis.

To find the angle a line makes with the positive x-axis, we often use trigonometric functions. For a point given on the plane, the angle can be found based off its coordinates, and more specifically, the tangent function can be especially useful in this context.

For the point \((0, 7)\), it's directly on the y-axis. This results in a vertical line, which corresponds to an angle of \(90^\circ\) relative to the positive x-axis. Therefore, the angle in standard position for this point is simply \(90^\circ\), showcasing how certain positions correspond to known angles.

Tangent Function

The Tangent Function is a trigonometric function that plays a vital role in finding angles in coordinate geometry. It's defined as the ratio of the opposite side to the adjacent side in a right triangle, often written as:\[\tan(\theta) = \frac{y}{x}\]where \(\theta\) is the angle in question.

When finding angles in standard position, this function aids in determining the slope or inclination of a line passing through the origin and a given point on the plane. However, an important exception exists when the x-coordinate is zero because division by zero is undefined. This special case occurs when the line is vertical, leading to a tangent value that's undefined.

• For horizontal lines, \(y = 0\) leads to \(\theta = 0^\circ\).
• For vertical lines, like in our example with \(x = 0\), the angle is \(90^\circ\).

This understanding of the tangent function is essential for working through coordinate geometry problems where determining angles is required.