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Sagot :
Para determinar si dos ángulos son coterminales, debemos ver si, después de sumar o restar múltiplos enteros de [tex]\(360^{\circ}\)[/tex] (en el caso de ángulos en grados) o [tex]\(2\pi\)[/tex] (en el caso de ángulos en radianes), ambos ángulos se reducen al mismo valor. Vamos a realizar estos cálculos uno por uno para cada par de ángulos.
### (a) [tex]\(1000^{\circ}\)[/tex] y [tex]\(280^{\circ}\)[/tex]
Primero, reducimos [tex]\(1000^{\circ}\)[/tex]:
[tex]\[ 1000^{\circ} \mod 360^{\circ} = 1000 - 2(360) = 1000 - 720 = 280^{\circ} \][/tex]
Luego comparamos con [tex]\(280^{\circ}\)[/tex]:
[tex]\[ 280^{\circ} = 280^{\circ} \][/tex]
Dado que ambos se reducen al mismo valor, [tex]\(1000^{\circ}\)[/tex] y [tex]\(280^{\circ}\)[/tex] son coterminales.
### (b) [tex]\(135^{\circ}\)[/tex] y [tex]\(-225^{\circ}\)[/tex]
Primero, reducimos [tex]\(135^{\circ}\)[/tex]:
[tex]\[ 135^{\circ} \mod 360^{\circ} = 135^{\circ} \][/tex]
Luego, reducimos [tex]\(-225^{\circ}\)[/tex]:
[tex]\[ -225^{\circ} \mod 360^{\circ} = -225 + 360 = 135^{\circ} \][/tex]
Ambos valores se reducen a [tex]\(135^{\circ}\)[/tex], por lo que [tex]\(135^{\circ}\)[/tex] y [tex]\(-225^{\circ}\)[/tex] son coterminales.
### (c) [tex]\(\frac{2 \pi}{5}\)[/tex] rad y [tex]\(-\frac{2 \pi}{5}\)[/tex] rad
Comparamos directamente:
[tex]\[ \frac{2 \pi}{5} \mod 2\pi = \frac{2 \pi}{5} \][/tex]
Para [tex]\(-\frac{2 \pi}{5}\)[/tex]:
[tex]\[ -\frac{2 \pi}{5} \mod 2\pi = -\frac{2 \pi}{5} + 2\pi = \frac{8 \pi}{5} \][/tex]
Dado que [tex]\(\frac{2\pi}{5} \neq \frac{8\pi}{5}\)[/tex], los ángulos no son coterminales.
### (d) [tex]\(\frac{5 \pi}{4}\)[/tex] rad y [tex]\(-\frac{3 \pi}{4}\)[/tex] rad
Primero, reducimos [tex]\(\frac{5 \pi}{4}\)[/tex]:
[tex]\[ \frac{5 \pi}{4} \mod 2\pi = \frac{5 \pi}{4} \][/tex]
Para [tex]\(-\frac{3 \pi}{4}\)[/tex]:
[tex]\[ -\frac{3 \pi}{4} \mod 2\pi = -\frac{3 \pi}{4} + 2\pi = \frac{5 \pi}{4} \][/tex]
Dado que ambos se reducen a [tex]\(\frac{5 \pi}{4}\)[/tex], son coterminales.
### (e) [tex]\(30^{\circ}\)[/tex] y [tex]\(410^{\circ}\)[/tex]
Primero, reducimos [tex]\(410^{\circ}\)[/tex]:
[tex]\[ 410^{\circ} \mod 360^{\circ} = 410 - 360 = 50^{\circ} \][/tex]
Para [tex]\(30^{\circ}\)[/tex]:
[tex]\[ 30^{\circ} = 30^{\circ} \][/tex]
Dado que [tex]\(50^{\circ} \neq 30^{\circ}\)[/tex], no son coterminales.
### (f) [tex]\(60^{\circ}\)[/tex] y [tex]\(-420^{\circ}\)[/tex]
Primero, reducimos [tex]\(60^{\circ}\)[/tex]:
[tex]\[ 60^{\circ} = 60^{\circ} \][/tex]
Para [tex]\(-420^{\circ}\)[/tex]:
[tex]\[ -420^{\circ} \mod 360^{\circ} = -420 + 360 = -60 \mod 360 = 300^{\circ} \][/tex]
Dado que [tex]\(300^{\circ} \neq 60^{\circ}\)[/tex], no son coterminales.
## Resultados
a. [tex]\(1000^{\circ}\)[/tex] y [tex]\(280^{\circ}\)[/tex] son coterminales.
b. [tex]\(135^{\circ}\)[/tex] y [tex]\(-225^{\circ}\)[/tex] son coterminales.
c. [tex]\(\frac{2 \pi}{5}\)[/tex] rad y [tex]\(-\frac{2 \pi}{5}\)[/tex] rad no son coterminales.
d. [tex]\(\frac{5 \pi}{4}\)[/tex] rad y [tex]\(-\frac{3 \pi}{4}\)[/tex] rad son coterminales.
e. [tex]\(30^{\circ}\)[/tex] y [tex]\(410^{\circ}\)[/tex] no son coterminales.
f. [tex]\(60^{\circ}\)[/tex] y [tex]\(-420^{\circ}\)[/tex] no son coterminales.
### (a) [tex]\(1000^{\circ}\)[/tex] y [tex]\(280^{\circ}\)[/tex]
Primero, reducimos [tex]\(1000^{\circ}\)[/tex]:
[tex]\[ 1000^{\circ} \mod 360^{\circ} = 1000 - 2(360) = 1000 - 720 = 280^{\circ} \][/tex]
Luego comparamos con [tex]\(280^{\circ}\)[/tex]:
[tex]\[ 280^{\circ} = 280^{\circ} \][/tex]
Dado que ambos se reducen al mismo valor, [tex]\(1000^{\circ}\)[/tex] y [tex]\(280^{\circ}\)[/tex] son coterminales.
### (b) [tex]\(135^{\circ}\)[/tex] y [tex]\(-225^{\circ}\)[/tex]
Primero, reducimos [tex]\(135^{\circ}\)[/tex]:
[tex]\[ 135^{\circ} \mod 360^{\circ} = 135^{\circ} \][/tex]
Luego, reducimos [tex]\(-225^{\circ}\)[/tex]:
[tex]\[ -225^{\circ} \mod 360^{\circ} = -225 + 360 = 135^{\circ} \][/tex]
Ambos valores se reducen a [tex]\(135^{\circ}\)[/tex], por lo que [tex]\(135^{\circ}\)[/tex] y [tex]\(-225^{\circ}\)[/tex] son coterminales.
### (c) [tex]\(\frac{2 \pi}{5}\)[/tex] rad y [tex]\(-\frac{2 \pi}{5}\)[/tex] rad
Comparamos directamente:
[tex]\[ \frac{2 \pi}{5} \mod 2\pi = \frac{2 \pi}{5} \][/tex]
Para [tex]\(-\frac{2 \pi}{5}\)[/tex]:
[tex]\[ -\frac{2 \pi}{5} \mod 2\pi = -\frac{2 \pi}{5} + 2\pi = \frac{8 \pi}{5} \][/tex]
Dado que [tex]\(\frac{2\pi}{5} \neq \frac{8\pi}{5}\)[/tex], los ángulos no son coterminales.
### (d) [tex]\(\frac{5 \pi}{4}\)[/tex] rad y [tex]\(-\frac{3 \pi}{4}\)[/tex] rad
Primero, reducimos [tex]\(\frac{5 \pi}{4}\)[/tex]:
[tex]\[ \frac{5 \pi}{4} \mod 2\pi = \frac{5 \pi}{4} \][/tex]
Para [tex]\(-\frac{3 \pi}{4}\)[/tex]:
[tex]\[ -\frac{3 \pi}{4} \mod 2\pi = -\frac{3 \pi}{4} + 2\pi = \frac{5 \pi}{4} \][/tex]
Dado que ambos se reducen a [tex]\(\frac{5 \pi}{4}\)[/tex], son coterminales.
### (e) [tex]\(30^{\circ}\)[/tex] y [tex]\(410^{\circ}\)[/tex]
Primero, reducimos [tex]\(410^{\circ}\)[/tex]:
[tex]\[ 410^{\circ} \mod 360^{\circ} = 410 - 360 = 50^{\circ} \][/tex]
Para [tex]\(30^{\circ}\)[/tex]:
[tex]\[ 30^{\circ} = 30^{\circ} \][/tex]
Dado que [tex]\(50^{\circ} \neq 30^{\circ}\)[/tex], no son coterminales.
### (f) [tex]\(60^{\circ}\)[/tex] y [tex]\(-420^{\circ}\)[/tex]
Primero, reducimos [tex]\(60^{\circ}\)[/tex]:
[tex]\[ 60^{\circ} = 60^{\circ} \][/tex]
Para [tex]\(-420^{\circ}\)[/tex]:
[tex]\[ -420^{\circ} \mod 360^{\circ} = -420 + 360 = -60 \mod 360 = 300^{\circ} \][/tex]
Dado que [tex]\(300^{\circ} \neq 60^{\circ}\)[/tex], no son coterminales.
## Resultados
a. [tex]\(1000^{\circ}\)[/tex] y [tex]\(280^{\circ}\)[/tex] son coterminales.
b. [tex]\(135^{\circ}\)[/tex] y [tex]\(-225^{\circ}\)[/tex] son coterminales.
c. [tex]\(\frac{2 \pi}{5}\)[/tex] rad y [tex]\(-\frac{2 \pi}{5}\)[/tex] rad no son coterminales.
d. [tex]\(\frac{5 \pi}{4}\)[/tex] rad y [tex]\(-\frac{3 \pi}{4}\)[/tex] rad son coterminales.
e. [tex]\(30^{\circ}\)[/tex] y [tex]\(410^{\circ}\)[/tex] no son coterminales.
f. [tex]\(60^{\circ}\)[/tex] y [tex]\(-420^{\circ}\)[/tex] no son coterminales.
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