Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(250^{\circ}\)

Step-by-Step Solution

Answer

The reference angle of \(250^{\circ}\) is \(70^{\circ}\).

Step 1: Identify and Sketch the Angle in Standard Position

To draw \(250^{\circ}\) in standard position, start by imagining a coordinate plane with the origin at the center. A standard angle begins at the positive \(x\)-axis. Measure counterclockwise from the positive \(x\)-axis to plot \(250^{\circ}\). The terminal side will fall in the third quadrant.

Step 2: Determine the Reference Angle

The reference angle is the acute angle (less than \(90^{\circ}\)) that the terminal side makes with the \(x\)-axis. In the third quadrant, the reference angle is calculated by subtracting \(180^{\circ}\) from the given angle. So our reference angle \(\theta_\text{ref} = 250^{\circ} - 180^{\circ} = 70^{\circ}\).

Step 3: Draw the Reference Angle

Draw the reference angle by starting from the \(x\)-axis in the third quadrant and measuring \(70^{\circ}\) in a clockwise direction towards the terminal side of the original angle from the \(x\)-axis.

Step 4: Draw the Reference Angle in Standard Position

To draw the \(70^{\circ}\) reference angle in standard position, start from the positive \(x\)-axis and measure \(70^{\circ}\) counterclockwise in the first quadrant. The terminal side of this angle should lie between the positive \(x\)-axis and the positive \(y\)-axis.

Step 5: State the Measure of the Reference Angle

The measure of the reference angle is \(70^{\circ}\). This is the acute angle between the terminal side of the original angle and the \(x\)-axis.

Key Concepts

Reference Angle

A reference angle is a crucial concept in trigonometry. It is the smallest angle that the terminal side of a given angle makes with the x-axis, always measured as a positive acute angle (less than 90°). It's helpful to visualize the reference angle as the bridge back to the x-axis from the terminal side. Here's why reference angles are important:

• They simplify calculations by allowing us to use well-known trigonometric values of acute angles.
• Reference angles help in determining the trigonometric function values of angles located in various quadrants.

To find the reference angle for an angle located in any quadrant other than the first, we subtract the necessary degrees to reach back to the x-axis. For example, an angle in the third quadrant like 250° would have a reference angle of 70°, because you subtract 180° from 250°.

Standard Position

An angle in standard position is always set on a coordinate plane with its vertex at the origin (0,0) and its initial side positioned along the positive x-axis. This starting point offers a consistent frame of reference for measuring angles counterclockwise around the origin. Why use standard position?

• It provides a universal method for measuring and discussing angles.
• This consistency makes it easier to visualize and solve trigonometric problems.

To visualize a given degree measure, imagine starting at the positive x-axis and rotating counterclockwise up to the given degree. For instance, to plot 250°, start from the positive x-axis and turn counterclockwise, landing in the third quadrant.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface made up of a horizontal x-axis and a vertical y-axis intersecting at the origin. This setup not only helps in defining the concept of angles but also forms the canvas upon which trigonometry problems are visualized. Key features:

• The x-axis runs from negative to positive horizontally, defining the direction for drawing angles in standard position.
• The y-axis runs vertically, helping to determine in which quadrant an angle lands.
• Four quadrants are defined, each holding significance based on where the terminal side of an angle rests.

The coordinate plane allows angles to be described not just by their measure but by the direction and rotation from the origin, making it indispensable in trigonometric calculations.

Terminal Side

The terminal side of an angle in the context of trigonometry is the ray that forms the endpoint of the angle's rotation from its initial position along the positive x-axis, when the angle is in standard position. Properties and importance:

• It defines the angle's placement within the coordinate plane's quadrants.
• The terminal side's intersection with the x-axis helps determine the reference angle.

For the angle measuring 250°, the terminal side falls in the third quadrant. This implies a counterclockwise rotation, passing the negative y-axis and making it easy to identify directly where the reference angle lies. This side, combined with knowledge of the coordinate plane, aids in understanding the full scope of angular positions and their trigonometric implications.