Question

question 21 (1 point)
the co - terminal angles to an angleθ may be determined using the expression
a) θ + 360°×n,n ∈ i
b) θ - 360°×n,n ∈ r
c) θ - 180°×n,n ∈ i
d) θ + 180°×n,n ∈ i

Explanation:

Co - terminal angles are angles that have the same initial and terminal sides. A full rotation is \(360^{\circ}\). When we add or subtract multiples of \(360^{\circ}\) (where \(n\) is an integer, \(n\in I\)) to an angle \(\theta\), we get co - terminal angles.
For example, if \(\theta = 30^{\circ}\) and \(n = 1\), then \(\theta+360^{\circ}\times n=30^{\circ}+ 360^{\circ}=390^{\circ}\), and \(30^{\circ}\) and \(390^{\circ}\) are co - terminal. If \(n=-1\), then \(\theta + 360^{\circ}\times n=30^{\circ}-360^{\circ}=- 330^{\circ}\), and \(30^{\circ}\) and \(-330^{\circ}\) are also co - terminal.

Now let's analyze each option:

• Option (a): \(\theta + 360^{\circ}\times n,n\in I\). Since adding or subtracting integer multiples (\(n\in I\)) of \(360^{\circ}\) to \(\theta\) gives co - terminal angles (when \(n\) is positive, we rotate counter - clockwise; when \(n\) is negative, we rotate clockwise), this is the correct formula.
• Option (b): \(n\in\mathbb{R}\) (where \(\mathbb{R}\) is the set of real numbers) is incorrect because we need to add or subtract integer multiples of \(360^{\circ}\) (not any real number multiple) to get co - terminal angles.
• Option (c): \(180^{\circ}\) is a half - rotation. Angles like \(\theta\) and \(\theta\pm180^{\circ}\) are not co - terminal (they are supplementary in some cases). For example, if \(\theta = 30^{\circ}\), \(\theta-180^{\circ}=-150^{\circ}\), and \(30^{\circ}\) and \(-150^{\circ}\) are not co - terminal.
• Option (d): Similar to option (c), \(180^{\circ}\) is not the measure of a full rotation. So, adding or subtracting integer multiples of \(180^{\circ}\) to \(\theta\) does not give co - terminal angles.

Answer:

A. \(\theta + 360^{\circ}\times n,n\in I\)