91Ó°ÊÓ

Angles in standard position are measured from a fixed direction, usually from the positive x-axis on a coordinate plane. This is an important concept in trigonometry. Here, the vertex of the angle is at the origin, while the initial side lies along the x-axis. The terminal side is the side where the angle ends after the rotation.
Understanding this helps in visualizing how angles are positioned and makes it easier to work with coterminal angles. For example, - If an angle opens counterclockwise, it is considered a positive angle. - If it opens clockwise, it is a negative angle.
This position serves as a reference point for calculating other angles, including those that are coterminal.

Positive angles are measured by the rotation from the initial side to the terminal side counterclockwise. This is generally the default direction for measuring angles. To find two positive angles that are coterminal with a given angle, like \(-45^{\circ}\), you simply add full rotations, or multiples of \(360^{\circ}\).

• Start by adding \(360^{\circ}\) to the original angle: \(-45^{\circ} + 360^{\circ} = 315^{\circ}\).
• Then, add another \(360^{\circ}\) to get the next coterminal angle: \(315^{\circ} + 360^{\circ} = 675^{\circ}\).

These calculations show how some positive angles share the same terminal side although they go through differentamounts of rotation.

Negative angles are determined by rotating clockwise from the initial side. These angles can be a bit tricky because we naturally think in counterclockwise terms. For the purpose of finding angles coterminal with \(-45^{\circ}\), you'd subtract full circles of \(360^{\circ}\).

• First, subtract \(360^{\circ}\) from the original angle: \(-45^{\circ} - 360^{\circ} = -405^{\circ}\).
• Then, subtract once more to discover a second negative coterminal angle: \(-405^{\circ} - 360^{\circ} = -765^{\circ}\).

These steps show negative coterminal angles by linking them to a clockwise rotation that ends in the same position as the initial angle.

A full rotation is equivalent to \(360^{\circ}\). This is the key to understanding how coterminal angles work. By adding or subtracting full rotations, you create angles that look the same, as they all share the same terminal side.
A full rotation forms a complete circle, bringing you back to the starting point on the coordinate plane. This concept is useful in both identifying and generating coterminal angles. For instance:

• Adding \(360^{\circ}\) represents one full counterclockwise rotation, moving back to the same terminal position without changing direction.
• Subtracting \(360^{\circ}\) does the same, but in a clockwise manner, proving that negative and positive angles can share the same result through different paths.

Through full rotations, you can calculate an infinite number of angles that are coterminal, simply by continuing to add or subtract \(360^{\circ}\).