Problem 21 - 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. \((24,7)\)

Step-by-Step Solution

Answer

The distance from the origin is 25, and the angle \(\theta\) is \(16°\).

Step 1: Identify the Point Coordinates

Given point coordinates are \((x, y) = (24, 7)\). These represent the point in a two-dimensional coordinate plane.

Step 2: Understand the Distance Formula

The distance of a point \((x, y)\) from the origin \((0, 0)\) can be calculated using the distance formula: \[D = \sqrt{x^2 + y^2}\] where \(x\) and \(y\) are the coordinates of the point.

Step 3: Calculate the Distance

Substitute \(x = 24\) and \(y = 7\) into the distance formula: \[D = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25\]So, the distance from the origin is 25.

Step 4: Understand the Angle Calculation

To find the angle in standard position, use the tangent function where \( \tan(\theta) = \frac{y}{x} \). This relates the y-coordinate and x-coordinate of the point to the angle \(\theta\) in trigonometry.

Step 5: Calculate the Angle

Substitute \(y = 7\) and \(x = 24\) into the tangent function to find \(\theta\): \[\theta = \arctan\left(\frac{7}{24}\right)\]Calculate \(\theta\) using a calculator to find the angle in degrees: \(\theta \approx 16.26°\).Round to the nearest degree: \(\theta \approx 16°\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics translates geometric problems into algebraic equations, making them easier to solve. In a two-dimensional plane, we refer to the position of any point using an ordered pair

• The first value in the pair is called the x-coordinate or abscissa.
• The second value is the y-coordinate or ordinate.

For example, with our given point \((24, 7)\), the x-coordinate is 24, and the y-coordinate is 7.
Using coordinate geometry, we can easily solve various problems such as finding distances between points, angles between lines, and the equations of lines or circles.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry, used to calculate the distance between two points in a plane.
It's derived from the Pythagorean theorem, which you might remember from your earlier studies.
To find the distance from a point \((x, y)\) to the origin \((0, 0)\), you use the formula:\[D = \sqrt{x^2 + y^2}\]This equation calculates the hypotenuse of a right triangle, which is the distance from the origin to the point.
When we substitute \(x = 24\) and \(y = 7\) into the formula, we find that \(D = \sqrt{576 + 49} = \sqrt{625} = 25\).
Thus, the point \((24, 7)\) is 25 units away from the origin.

Trigonometric Functions

Trigonometry deals with the relationships between the angles and sides of triangles.
One of the key functions used in trigonometry is the tangent function, represented as \(\tan(\theta)\), where \(\theta\) is an angle.
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
For a point \((x, y)\) in the coordinate plane, the tangent can be calculated as:\[\tan(\theta) = \frac{y}{x}\]Using this formula, we can determine the angle given specific coordinates.
For example, with \(x = 24\) and \(y = 7\), \[\tan(\theta) = \frac{7}{24}\].This ratio helps us figure out the angle at the origin that forms with the positive x-axis and through the point.

Standard Position Angle

Understanding a standard position angle is crucial in studying angles on the coordinate plane. An angle is said to be in standard position if its vertex is at the origin and its initial side is along the positive x-axis.
The terminal side of the angle is the side that moves to measure the angle.
When calculating the angle whose terminal side passes through a given point \((x, y)\), you use the inverse tangent function:\[\theta = \arctan\left(\frac{y}{x}\right)\]This function helps find the measure of \(\theta\) in degrees.
For our example, substituting \(y = 7\) and \(x = 24\), we compute:\[\theta = \arctan\left(\frac{7}{24}\right)\].Using a calculator, we get \(\theta \approx 16.26°\).After rounding to the nearest degree, the angle becomes \(\theta \approx 16°\), indicating this angle from the x-axis to the line passing through \((24, 7)\).