Question

1. which expression represents the measures of all the angles coterminal with 301° in standard position?

a. 301° + k180°, k ∈ z
b. 59° + k360°, k ∈ z
c. 301° + k360°, k ∈ r
d. 301° + k360°, k ∈ z

Explanation:

Step1: Recall Coterminal Angles Definition

Coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles with a given angle \(\theta\) in degrees, we add or subtract multiples of \(360^\circ\) (since a full rotation is \(360^\circ\)). The general formula for all coterminal angles with \(\theta\) is \(\theta + k\cdot360^\circ\), where \(k\in\mathbb{Z}\) (integers, positive or negative).

Step2: Analyze Each Option

• Option A: \(301^\circ + k\cdot180^\circ\), \(k\in\mathbb{Z}\). This would give angles that are half - rotations apart (since \(180^\circ\) is a half - rotation), not full - rotations. So these are not coterminal with \(301^\circ\) in the sense of sharing the same terminal side after full rotations.
• Option B: \(59^\circ + k\cdot360^\circ\), \(k\in\mathbb{Z}\). The angle \(59^\circ\) is not equal to \(301^\circ\), so this formula does not represent angles coterminal with \(301^\circ\).
• Option C: \(301^\circ + k\cdot360^\circ\), \(k\in\mathbb{R}\). \(k\) should be an integer because we are adding or subtracting whole number of full rotations. If \(k\) is a real number, it doesn't make sense for the number of rotations (we can't have a fraction of a rotation in the context of coterminal angles defined by full rotations).
• Option D: \(301^\circ + k\cdot360^\circ\), \(k\in\mathbb{Z}\). This follows the formula for coterminal angles, where we add or subtract (when \(k\) is negative) multiples of \(360^\circ\) to \(301^\circ\), and \(k\) is an integer to represent whole number of full rotations.

Answer:

D. \(301^{\circ}+k360^{\circ},k\in Z\)