Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-7,\) draw each angle in standard position. $$ 540^{\circ} $$

Step-by-Step Solution

Answer

540° corresponds to an angle of 180° in standard position.

Step 1: Understanding Standard Position

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

Step 2: Determine Number of Full Rotations

Since a full rotation is 360°, we can determine how many full rotations 540° involves by dividing 540° by 360°: \\[\text{Full rotations} = \frac{540}{360} = 1.5.\]

Step 3: Calculate Equivalent Angle within One Rotation

To find the equivalent angle within one rotation (less than 360°), take the remainder of 540° divided by 360°: \\[540° \mod 360° = 180°.\]

Step 4: Draw the Angle

Start from the positive x-axis, move counterclockwise for one full rotation (360°), and then move an additional 180° around the circle. This results in the terminal side lying on the negative x-axis.

Key Concepts

Full Rotations

When dealing with angles in standard position, it's crucial to understand the concept of full rotations. A full rotation is when an angle turns around the entire circle, which is exactly 360 degrees.
- This means any angle that measures exactly 360 degrees will have completed one full rotation. - Angles can have more than one full rotation if they exceed 360 degrees. For example, a 720-degree angle completes two full rotations since 720 divided by 360 equals 2.
In the exercise, the angle given is 540 degrees. To find out how many full rotations 540 degrees consists of, you divide by 360 degrees. This gives 1.5, which means after one full 360-degree rotation, there is an additional half rotation completed.

Equivalent Angle Within One Rotation

After completing any full rotations, angles that are larger than 360 degrees can be simplified by calculating their equivalent angle within one rotation. This equivalent angle helps to determine the same position on the coordinate plane, but represented by a smaller, easier-to-manage angle.
- To find it, you take the original angle and remove the full rotations by using the modulus operation. For angles measured in degrees, this involves finding the remainder when dividing by 360.
In our example exercise, 540 degrees modulo 360 degrees results in an equivalent angle of 180 degrees. This means that, despite starting with a larger angle, it can be represented just as accurately within the coordinate plane by considering it as 180 degrees.

Terminal Side on Coordinate Plane

The terminal side of an angle in standard position is where the angle stops rotating. It is crucial for visually understanding the angle's position within the coordinate system.
- To draw this, start from the initial side located on the positive x-axis. - Rotate counterclockwise for positive angles or clockwise for negative angles.
From the supplied example, after completing one full 360-degree rotation, there are still 180 degrees to cover. So, from the positive x-axis, the angle turns another 180 degrees counterclockwise, landing on the negative x-axis. This positioning helps identify both the direction and final resting place of an angle in the coordinate plane.