Problem 16 - 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. \((-5,12)\)

Step-by-Step Solution

Answer

a. The distance from the origin is 13. b. The angle is approximately 113°.

Step 1: Identify Coordinates

The coordinates of the given point are (-5, 12). Here, x = -5 and y = 12.

Step 2: Use Distance Formula

The distance of a point (x, y) from the origin (0, 0) is given by: \(d = \sqrt{x^2 + y^2}\). Substitute x = -5 and y = 12 into the formula: \(d = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\). Thus, the distance from the origin is 13 units.

Step 3: Use Trigonometry to Find Angle

The angle θ in standard position with terminal side passing through (-5, 12) can be calculated using the tangent function: \(\tan(θ) = \frac{y}{x} = \frac{12}{-5} = -2.4\).

Step 4: Calculate Angle Using Arctangent

Find the angle θ using the inverse tangent function: \(θ = \arctan(-2.4)\). This calculates to approximately θ ≈ -67.38^ ext{o}. However, since the point (-5, 12) is in the second quadrant, adjust the angle: θ ≈ 180 + (-67.38) = 112.62^ ext{o}.

Step 5: Round to Nearest Degree

Round 112.62^ ext{o} to the nearest degree, resulting in 113^ ext{o}.

Key Concepts

Distance Formula

To find the distance from the origin to a point in the coordinate plane, use the Distance Formula. This is especially useful in helping us understand spatial relationships and measurements in both mathematics and real-life contexts like physics and engineering.
The Distance Formula is expressed as:

• \(d = \sqrt{x^2 + y^2}\)

Here, \(d\) is the distance, \(x\) is the x-coordinate, and \(y\) is the y-coordinate of the point. When you substitute the given coordinates into the formula, you effectively construct a right triangle, with the horizontal and vertical distances repsresenting the legs of the triangle.
Applying the Pythagorean theorem, you compute the hypotenuse, which is the distance from the origin. This makes understanding the concept rather intuitive because it's based on the classic right triangle method.

• Example for a point (-5, 12): \[ d = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

The calculated distance, therefore, is 13 units from the origin.

Trigonometry

Trigonometry helps us link angles to the lengths of sides in right triangles. It is essential for understanding geometrical relationships in various branches of science and engineering. When working with coordinates, the primary trigonometric function often applied is the tangent function. It relates the angle to the opposite and adjacent sides of a right triangle:

• \(\tan(θ) = \frac{y}{x}\)

Where \(θ\) is the angle in standard position whose terminal side contains the point situated relative to the origin, \(y\) is the vertical distance, and \(x\) is the horizontal distance.
This relationship provides a straightforward method for calculating the angle when given the coordinates. It works even when the coordinates result in negative values, as long as you appropriately adjust for the quadrant in which the angle lies.

• Example with point (-5, 12): \[ \tan(θ) = \frac{12}{-5} = -2.4 \]

Inverse Tangent Function

The Inverse Tangent Function, or Arctangent, is used to find an angle when you know the tangent value. It helps determine the angle whose tangent is the given number and is particularly useful when working with points on the coordinate plane.
The function is written as:

• \(θ = \arctan\left(\frac{y}{x}\right)\)

This step is crucial because direct calculation using a calculator may result in an angle that needs adjustment based on its position in the coordinate system—adjustments arise since tangent values repeat every 180 degrees. When analyzing a point like (-5, 12), you calculate the inverse tangent value:

• \(θ = \arctan(-2.4) \approx -67.38^\circ\)

Since (-5, 12) lies in the second quadrant where the angle range is from 90 to 180 degrees, adjust the angle found to fit the correct quadrant:

• \(θ \approx 180 + (-67.38) = 112.62^\circ\)

Round this final angle to the nearest degree to find \(113^\circ\), providing a precise measure of the angular position.