Problem 33 - Algebra and Trigonometry

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 70^{\circ}, \quad 430^{\circ} $$

Step-by-Step Solution

Answer

Yes, the angles are coterminal (70° and 430°).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles in standard position (with the vertex at the origin and the initial side along the positive x-axis) that have the same terminal side. Coterminal angles can be determined by adding or subtracting full rotations of 360 degrees to one of the angles.

Step 2: Determine Extra Rotations

To determine if 430° is coterminal with 70°, subtract 360° from 430° to see if it can be reduced to 70° or another angle coterminal with 70°. 430° - 360° = 70°.

Step 3: Compare Results

The result from the previous step is 70°. Since 70° is the angle we are comparing to, both 70° and 430° have the same terminal side, making them coterminal angles.

Key Concepts

Angle Measurement

Understanding angles is crucial in geometry. In simple terms, an angle is formed by two rays (or sides) that have a common endpoint, known as the vertex. Angles are measured in degrees, which quantify how much one ray rotates around the vertex to reach the position of the other ray.

• One complete revolution around a circle is measured as 360 degrees.
• When we talk about angle measurement, it is important to understand how and why angles are expressed in certain numerical values.

By using degrees, we can easily express an angle's rotation and determine how much an angle turns. In problems involving angle measurement, you'll often need to determine whether two angles share the same position with the help of coterminality.

Standard Position

When we refer to an angle's standard position, we are placing that angle in a common framework that helps in identifying its characteristics. In the Cartesian coordinate system, the standard position of an angle has two main attributes:

• The vertex is located at the origin of the coordinate system.
• The initial side coincides with the positive x-axis.

Positioning angles this way helps in comparing and contrasting different angles. When two angles share the same terminal side in the standard position, we identify them as coterminal angles. This concept helps us see how an angle's measure can change by full rotations, yet still end up in the same place geometrically.

Full Rotations

To fully understand coterminal angles, we must grasp the idea of full rotations. A full rotation is a complete 360-degree turn, bringing a ray back to its starting point. This concept is central when determining if two angles are coterminal.

• Adding or subtracting full rotations (multiples of 360 degrees) from an angle doesn't change its terminal side's position.
• This principle helps find new representations of an angle that structurally align along the same terminal side.

By repeatedly adding or subtracting 360 degrees from an angle, we can generate many coterminal angles. For example, subtracting 360 degrees from 430 degrees results in 70 degrees, demonstrating that these angles are coterminal, even though their numerical measures differ.