Problem 6 - 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. \(-45^{\circ}\)

Step-by-Step Solution

Answer

Reference angle is \(45^{\circ}\).

Step 1: Standard Position of the Angle

Start by drawing the angle \(-45^{\circ}\) in the standard position. The standard position means that the vertex of the angle is at the origin of the coordinate plane, and its initial side lies along the positive \(x\)-axis. From there, rotate clockwise since the angle is negative. The terminal side will cross into the fourth quadrant.

Step 2: Identify the Reference Angle

The reference angle is the smallest angle that the terminal side makes with the \(x\)-axis. Since the terminal side of the angle \(-45^{\circ}\) is in the fourth quadrant, the reference angle is formed by the terminal side and the positive \(x\)-axis.

Step 3: Find the Measure of the Reference Angle

Calculate the reference angle by taking the absolute value of the original angle. Since \(-45^{\circ}\) is negative and less than \(0^{\circ}\), the reference angle will simply be \(|-45^{\circ}| = 45^{\circ}\).

Step 4: Draw the Reference Angle in Standard Position

Now, draw the reference angle \(45^{\circ}\) in standard position. Place the vertex at the origin with the initial side on the positive \(x\)-axis, and then draw the final side rotating upward (counterclockwise) by \(45^{\circ}\). This will show the reference angle in the first quadrant.

Key Concepts

Standard Position

Understanding angles in their standard position is crucial when studying geometry. An angle is said to be in standard position if its vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis. But what happens when an angle is negative? Negative angles are drawn by rotating the initial side clockwise. For instance, an angle of -45° would start on the positive x-axis and rotate clockwise into the fourth quadrant. Here's why the standard position is important:

• It gives a consistent starting point to measure angles.
• It helps in visualizing the angle's relation to the coordinate plane and quadrants.
• It ensures that everyone measures angles the same way.

Recognizing standard position leads us to better understand different types of angles and their relationships in a coordinate plane.

Angle Measurement

Angle measurement is more than just a number. It tells us how far one line revolves around another. It’s fundamental in understanding rotations and directions in geometry. There are different units for measuring angles, but the most common in geometry classes is degrees. Here's a quick run-through:

• One full rotation around a point is 360 degrees.
• Angles can be positive (counterclockwise) or negative (clockwise).
• Acute angles are less than 90°, right angles equal 90°, and obtuse angles are between 90° and 180°.

These distinctions are critical when comparing and working with angles. For example, a -45° angle shows a clockwise rotation, and its measurement helps find the reference angle, which is crucial for solving problems in standard position.

Coordinate Plane

A coordinate plane is the playground for graphing points, lines, and angles. Comprised of two perpendicular axes, the x-axis (horizontal) and y-axis (vertical), it divides the plane into four parts known as quadrants. The origin, where the axes meet, is the reference point for all measurements. Understanding the coordinate plane allows you to do several things:

• Plot points using (x, y) coordinates.
• Visualize geometric shapes and their relations to the axes.
• Identify the orientation of angles and locate their positions precisely.

With the coordinate plane, we can accurately position angles in standard form and analyze their direction and relationships in geometric problems.

Quadrants

Quadrants are one of the most important concepts on the coordinate plane. The plane is divided into four sections called quadrants, numbered I to IV, counterclockwise starting from the upper right. These sections help us know where a point or angle lies:

• Quadrant I has positive x and y values.
• Quadrant II has negative x but positive y values.
• Quadrant III allows both x and y to be negative.
• Quadrant IV has positive x but negative y values.

Knowing which quadrant an angle's terminal side falls into is key. It affects the direction of rotation and determination of the reference angle. For example, the -45° angle's terminal side falls in Quadrant IV, informing us about its properties and aiding in solving related problems.