Question

observa la imagen del ángulo $-270^{\circ}$ en posición estándar.

imagen de un sistema de ejes coordenados con un ángulo de $-270^{\circ}$ representado

¿cuál ángulo es coterminal con el ángulo $-270^{\circ}$?

$\bigcirc$ a. $270^{\circ}$

$\bigcirc$ b. $180^{\circ}$

$\bigcirc$ c. $-90^{\circ}$

$\bigcirc$ d. $-630^{\circ}$

Explanation:

Step1: Recall coterminal angle formula

Coterminal angles differ by multiples of \(360^\circ\). For an angle \(\theta\), coterminal angles are \(\theta + 360^\circ n\) where \(n\) is an integer.

Step2: Test each option

• Option A: \(270^\circ - (-270^\circ)=540^\circ\), not a multiple of \(360^\circ\).
• Option B: \(180^\circ - (-270^\circ)=450^\circ\), not a multiple of \(360^\circ\).
• Option C: \(-90^\circ - (-270^\circ)=180^\circ\), not a multiple of \(360^\circ\). Wait, maybe we should add \(360^\circ\) to \(-270^\circ\): \(-270^\circ+360^\circ = 90^\circ\)? No, wait, maybe I made a mistake. Wait, let's check option D: \(-630^\circ - (-270^\circ)= -360^\circ\), which is a multiple of \(360^\circ\) (\(n = - 1\) since \(-270^\circ+360^\circ(-1)=-630^\circ\)). Wait, no, wait the formula is \(\theta + 360n\). Let's check each angle:

For angle \(\alpha\) to be coterminal with \(-270^\circ\), \(\alpha=-270^\circ + 360^\circ n\), \(n\in\mathbb{Z}\).

• For \(n = - 1\): \(-270^\circ+360^\circ(-1)=-630^\circ\) (matches option D).

Wait, earlier mistake: when checking option D, \(-630^\circ=-270^\circ - 360^\circ\), so it's coterminal. Let's re - check:

Coterminal angles are angles that share the same terminal side. So, if we add or subtract \(360^\circ\) (or \(2\pi\) in radians) to an angle, we get a coterminal angle.

For \(-270^\circ\), let's find \(n\) such that \(-270^\circ+360^\circ n\) gives one of the options.

• For \(n=-1\): \(-270 + 360\times(-1)=-270 - 360=-630^\circ\) (option D).

Let's check other options:

• Option A: \(270^\circ=-270^\circ+360n\Rightarrow360n = 540\Rightarrow n = 1.5\) (not integer).
• Option B: \(180^\circ=-270^\circ+360n\Rightarrow360n = 450\Rightarrow n = 1.25\) (not integer).
• Option C: \(-90^\circ=-270^\circ+360n\Rightarrow360n = 180\Rightarrow n = 0.5\) (not integer).
• Option D: \(-630^\circ=-270^\circ+360n\Rightarrow360n=-360\Rightarrow n=-1\) (integer, so valid).

Answer:

D. \(-630^\circ\)