Problem 42 - Basic Technical Mathematics with Calculus

Question

Draw angles in standard position such that the terminal side passes through the given point. $$(-4,-2)$$

Step-by-Step Solution

Answer

The angle in standard position is approximately 206.6°.

Step 1: Understand the Coordinate Point

The point (-4, -2) is given in a coordinate plane, where the x-coordinate is -4, and the y-coordinate is -2.

Step 2: Plot the Point on the Coordinate Plane

Place the point (-4, -2) on a standard coordinate plane. This point lies in the third quadrant, where both x and y values are negative.

Step 3: Draw the Radius Vector

From the origin (0, 0), draw a straight line passing through the point (-4, -2). This line is the terminal side of the angle in standard position.

Step 4: Determine the Angle

In standard position, angles are measured counterclockwise from the positive x-axis. The terminal side calculated will form an angle with the horizontal axis. Use the inverse tangent function to find the angle \( \theta \) as follows: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) = \tan^{-1} \left( \frac{-2}{-4} \right) = \tan^{-1} \left( \frac{1}{2} \right) \approx 26.6^\circ \]Since the angle is in the third quadrant, add 180° to \( \theta \):\[ \theta_{final} = 180^\circ + 26.6^\circ = 206.6^\circ \]

Step 5: Draw the Angle

Start from the positive x-axis and move counterclockwise, marking an angle of approximately 206.6° until reaching the terminal side that passes through (-4, -2). This is the angle in standard position.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, helps to understand the spatial properties of points in a plane by using a coordinate system. In a coordinate plane, every point is defined by an ordered pair (x, y), where 'x' represents the horizontal distance and 'y' the vertical distance from the origin, which is the point (0, 0).
The exercise uses the point (-4, -2), meaning it is located 4 units left and 2 units down from the origin. Understanding positions on the coordinate plane is crucial since it allows us to determine which quadrant a point is located in and helps in visualizing geometric figures and angles.

Inverse Tangent Function

The inverse tangent function, denoted as \( \tan^{-1} \) or arctan, is a crucial tool in trigonometry used to find the angle whose tangent is a given number. The tangent of an angle in a right triangle is calculated as

• the ratio of the length of the opposite side to the length of the adjacent side.

When given a point (x, y), the function \( \tan^{-1} \left( \frac{y}{x} \right) \) helps us find the angle \( \theta \) formed with the x-axis.
Consider the exercise involving the point (-4, -2). Substituting into the formula gives: \( \theta = \tan^{-1} \left( \frac{-2}{-4} \right) = \tan^{-1} \left( \frac{1}{2} \right) \), about 26.6 degrees. This angle is taken with respect to the reference angle in standard position.

Standard Position of Angles

Angles are considered in standard position in coordinate geometry when their vertex is located at the origin, and their initial side lies along the positive x-axis. The measurement is taken by moving counterclockwise for positive angles and clockwise for negative angles.
This is why we say the angle is in standard position when its initial point starts from the positive x-axis and ends at the terminal side passing through the point of interest. For a point in the third quadrant, like (-4, -2), we find the reference angle using the inverse tangent first, which is about 26.6 degrees. Then, since it is in the third quadrant, 180 degrees is added, giving the final standard angle position as 206.6 degrees.

Quadrants in Geometry

The coordinate plane is divided into four quadrants which help in indicating the sign of the x and y coordinates for any given point. They are numbered counterclockwise as follows:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: x is negative, y is negative.
• Fourth Quadrant: x is positive, y is negative.

In the case of the exercise, the point (-4, -2) falls in the third quadrant, where both the x and y coordinates are negative. Understanding quadrants is essential in determining how the angle should be adjusted when converting from the inverse tangent calculation to the actual angle in the standard position.