Problem 37 - 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 26.57 degrees.

Step 1: Understand Standard Position

An angle is in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis.

Step 2: Identify the Quadrant

To determine in which quadrant the angle lies, consider the point (4,2). Since both x and y coordinates are positive, the point lands in the first quadrant.

Step 3: Calculate the Reference Angle

To find the angle, we use the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. Here, tan(θ) = 2/4 = 1/2.

Step 4: Use Inverse Tangent

Find the angle θ using the inverse tangent function: θ = tan^{-1}(1/2). Use a calculator to find that θ is approximately 26.57 degrees.

Step 5: Draw the Angle

Begin with the initial side on the positive x-axis. Rotate counterclockwise by approximately 26.57 degrees so that the terminal side passes through the point (4,2).

Step 6: Check Your Work

Verify that the angle is correct by ensuring that its tangent (opposite/adjacent) matches the given point, which confirms that θ approximately equals 26.57 degrees.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points by using an ordered pair of numbers known as coordinates. These coordinates indicate the location, with the first number corresponding to the x-axis and the second number to the y-axis.

In the context of angles in standard position, this plane is crucial as it helps us visualize and draw angles by providing a reference framework.

• The horizontal axis is referred to as the x-axis.
• The vertical axis is the y-axis.
• The point where these axes intersect is called the origin, typically denoted as (0,0).

The coordinate plane is divided into four sections, known as quadrants, which help in determining the direction and position of angles and points.

Reference Angle

A reference angle is the smallest angle that the terminal side makes with the x-axis. It is always a positive angle and helps in understanding how an angle interacts with the coordinate plane.

Reference angles are particularly useful when dealing with angles greater than 90 degrees or less than 0 degrees. They allow us to work with the simple geometric properties of right triangles, regardless of the quadrant in which the terminal side lies.

• A reference angle is always between 0 and 90 degrees.
• Even if an angle exceeds these values, a simpler, equivalent angle can be found within these limits.

This helps simplify calculations and make trigonometric functions easy to handle.

Inverse Tangent Function

The inverse tangent function, denoted as tan^{-1} or arctan, helps us find angles when given the tangent value. It is the reverse process of the tangent function, which is one of the primary trigonometric functions connected with right-angle triangles.

The tangent function itself is the ratio of the side opposite to the angle to the side adjacent to the angle (in a right triangle). When you know the ratio, the inverse tangent function allows you to figure out the angle in degrees or radians.

• The general form of the inverse tangent function is: \(\theta = \tan^{-1}(\text{value})\).
• When using a calculator, ensure that it's set to the correct mode (degrees or radians).

This function is especially useful when determining an angle from measurable lengths of triangle sides.

Quadrants of Coordinate Plane

The coordinate plane is divided into four quadrants, each containing different signs for the x and y coordinates. Knowing which quadrant a point is in can help determine the general direction and position of an angle in standard position.

The quadrants are numbered counterclockwise starting from the top right:

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

Understanding these quadrants is key to correctly plotting and identifying the position of angles, especially when the angle measures more than 90 degrees.