Problem 67 - Algebra 2

Question

Sketch each angle in standard position. $$ -85^{\circ} $$

Step-by-Step Solution

Answer

To sketch -85 degrees angle in standard position, start from the right of the x-axis (which is the initial side) and move 85 degrees in the clockwise direction (since the angle is negative). Draw the terminal side to complete the angle.

Step 1: Draw the initial side of the angle

Start by drawing the x and y axes on a piece of graph paper. Then draw a horizontal line starting from the origin (0, 0) extending to the right. This line serves as the initial side of the angle.

Step 2: Determine the direction and magnitude of the angle

Because the magnitude of the angle is negative, the direction of the angle is clockwise. Draw the terminal side of the angle 85 degrees clockwise from the initial side.

Step 3: Complete the angle

Complete the angle by drawing an arrow starting from the initial side and ending at the terminal side. This arrow represents the -85 degrees angle.

Key Concepts

Standard Position

In angle measurement, positioning plays a crucial role. The term "standard position" refers to placing an angle in a specific manner on a coordinate plane. Here's what this involves:

• The vertex of the angle is always at the origin (0, 0) of the graph, where the x-axis and y-axis intersect.
• The initial side of the angle always lies along the positive x-axis.

This setup helps in maintaining consistency and making calculations easier. Such positioning allows for a straightforward starting point for drawing and analyzing angles. When you place an angle in standard position, you can easily describe it using positive or negative values, which leads us to our next topic: negative angles.

Negative Angles

Negative angles might seem tricky at first, but they are simply angles that move in the opposite direction of positive angles.

• Positive angles are measured counterclockwise from the initial side of the angle.
• Negative angles, conversely, are measured clockwise from the initial side.

For instance, an angle of \(-85^{\circ}\) means you start from the x-axis and move 85 degrees in the clockwise direction. Visualizing this on a coordinate plane ensures that negative angles are just as intuitive as positive ones, simplifying tasks like sketching angles in standard position.

Graphing Angles

Graphing angles involves plotting them on the coordinate plane beginning from the standard position. Here's how this process unfolds:

• Start by drawing the x-axis and y-axis intersecting at the origin.
• Align the initial side of the angle with the positive x-axis.
• Determine the angle's direction and magnitude (positive or negative) to decide the direction of rotation.
• Locate the terminal side of the angle accordingly.

This visualization helps in understanding the angular measurements and is particularly useful for angles like \(-85^{\circ}\). By following these steps, graphing angles becomes a manageable task, aiding in better comprehension of their properties.

Initial Side and Terminal Side of the Angle

An angle is defined by its two sides: the initial side and the terminal side. Here's a closer look:

• The initial side is where the angle begins, typically lying on the positive x-axis when in standard position.
• The terminal side is where the angle ends after a certain rotation.

The direction to reach the terminal side can be clockwise or counterclockwise, depending on whether the angle is negative or positive. This understanding is essential while sketching and analyzing angles like \(-85^{\circ}\). Knowing both these sides allows for precise depiction and interpretation of angles within the coordinate plane.