Problem 36 - Algebra and Trigonometry

Question

\(31-36\) me measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 50^{\circ}, \quad 340^{\circ} $$

Step-by-Step Solution

Answer

The angles are not coterminal as their difference is not a multiple of \(360^{\circ}\).

Step 1: Understand Coterminal Angles

Coterminal angles have the same initial and terminal sides when drawn in standard position on the coordinate plane. To determine if angles are coterminal, they should differ by a multiple of a full rotation, i.e., \(360^{\circ}\) or \(2\pi\) radians.

Step 2: Calculate the Difference

Calculate the difference between the two given angles: \(340^{\circ} - 50^{\circ} = 290^{\circ}\).

Step 3: Check for Multiple of 360 Degrees

Check if the difference of \(290^{\circ}\) is a multiple of \(360^{\circ}\). A difference of \(290^{\circ}\) is not a multiple of \(360^{\circ}\), indicating that \(50^{\circ}\) and \(340^{\circ}\) are not coterminal.

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in geometry, crucial for understanding positions and rotations of angles on a coordinate plane. It determines how far around a circle the angle sweeps from the initial side to the terminal side.
Angles are typically measured in degrees or radians, which are the two major units. In degrees, a full circle is 360 degrees, while in radians, it is \(2\pi\) radians. To convert between degrees and radians, you can use the formulas:

• Degrees to radians: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)
• Radians to degrees: \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\)

Understanding these measurements is key to identifying coterminal angles, which share the same terminal side when plotted.
Grasping the concept of angle measurement helps in determining relationships and differences between angles, pertinent when solving trigonometry problems involving rotations and positions.

Standard Position

An angle is in standard position when its vertex is located at the origin of the coordinate plane, and its initial side lies along the positive x-axis. This consistent starting point allows for clear comparison and calculation of angles.
In the context of coterminal angles, standard position plays a critical role. When two angles have their initial sides on the positive x-axis and their terminal sides coincide, they are coterminal, despite possibly having different measures.
Angles in standard position make it easier to visualize their terminal sides and compare them directly. Therefore, when angles like \(50^{\circ}\) and \(340^{\circ}\) are analyzed, their positions help determine if their terminal sides overlap or how they relate to each other in terms of rotations.

Angle Difference

The difference between two angles provides insight into their relative positions after rotations. Calculating the difference helps in assessing whether two angles are coterminal.
For angles that are potentially coterminal, the difference should be an integer multiple of a full rotation, which is \(360^{\circ}\) or \(2\pi\) radians. This means:

• If the difference \(d\) between two angles satisfies \(d = n \times 360^{\circ}\), where \(n\) is an integer, the angles are coterminal.

In the task example, the difference between \(340^{\circ}\) and \(50^{\circ}\) is \(290^{\circ}\). Since \(290^{\circ}\) is not a multiple of \(360^{\circ}\), the angles do not share the same terminal side when in standard position.
Understanding angle difference clarifies whether different angles effectively represent the same position in circular motion, based on their rotational distances.