Problem 32 - Algebra and Trigonometry

Question

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ -30^{\circ}, \quad 330^{\circ} $$

Step-by-Step Solution

Answer

The angles -30° and 330° are coterminal.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. To determine if two angles are coterminal, we add or subtract multiples of 360° from one angle and check if we can obtain the other angle.

Step 2: Calculate the Equivalent Positive Angle for -30°

Since -30° is negative, we'll add 360° to it to convert it to its equivalent positive angle. \[-30° + 360° = 330°\]

Step 3: Compare the Calculated Angle with 330°

Now we compare the angle obtained in Step 2, which is 330°, with the given angle 330°. Since these two angles are equal, they are coterminal angles.

Key Concepts

Standard Position

When we talk about angles in the 'standard position,' we have a specific way of visualizing them on a coordinate plane. An angle is said to be in the standard position if its vertex is placed at the origin of the coordinate system with the initial side of the angle along the positive x-axis.
This setting helps in consistently identifying and comparing angles. By having a fixed initial side, we can easily determine the movement needed to form the angle, either in a clockwise or a counterclockwise direction.

• Angles moving counterclockwise from the positive x-axis are considered positive.
• Angles moving clockwise are considered negative.

Knowing this, it becomes easier to find the standard position of any angle, including handling those angles that are coterminal.

Angle Measures

Angle measures play a crucial role in understanding and identifying coterminal angles. Angles can be measured in degrees, and their positive or negative sign indicates the direction of rotation.

• Positive angles are those formed by counterclockwise rotation from the positive x-axis.
• Negative angles result from clockwise rotation.

For example, a positive angle of 330° means we rotate counterclockwise nearly a full circle, while a negative angle of -30° means a short clockwise rotation from the same initial side. This clockwise and counterclockwise characterization of angles helps make sense of their equivalence and relationships, which is essential to understanding ideas like coterminality.

Angle Equivalence

Two angles are considered equivalent if they terminate at the same position, even if their journeys to get there differ. Coterminal angles are a great example of this concept. They might have different numerical measures, but they end at the same final position.
To determine angle equivalence:

• Start with one angle.
• Add or subtract 360° to or from the angle to check for equivalence.
• Multiple rounds of adding or subtracting can be done to ensure all possibilities are checked.

The whole idea of angle equivalence underpins coterminal angles. In the exercise example, starting with -30° and adding 360° results in an equivalent angle of 330°, demonstrating that they are indeed coterminal, thus illustrating angle equivalence.