Answer:
Yes, they are coterminal.
Step-by-step explanation:
Coterminal angles are angles that are in multiples of -360 degrees or +360 degrees away from the initial angle.
Therefore, the easiest way to check if an angle is coterminal is set up an equation.
[tex]x + 117 = 477\\\\360 + 117 = 477\\\\477 = 477 \ \checkmark[/tex]
We get a true statement, which means that 117 is a coterminal angle of 477 and 477 is a coterminal angle of 117.
We can also determine the value in radians.
[tex]\displaystyle117 \times \frac{\pi}{180}\\\\\frac{117\pi}{180} = \frac{13\pi}{20}[/tex]
117° in radians is [tex]\displaystyle\frac{13\pi}{20}[/tex]. Coterminal angles for radians are multiples of [tex]2\pi[/tex], either negative or positive.
However, we also need to determine the value of 477 in radians too.
[tex]\displaystyle477\times\frac{\pi}{180}\\\\\frac{477\pi}{180} = \frac{53\pi}{20}[/tex]
So, if we add [tex]2\pi[/tex] to [tex]\frac{13\pi}{20}[/tex]:
[tex]\displaystyle\frac{13\pi}{20} + 2\pi\\\\\frac{13\pi}{20} + \frac{40\pi}{20} = \frac{53\pi}{20}[/tex]
Therefore, even in radians, 477° is an angle coterminal to 117°.
You are given 117°.
To find angles that are coterminal with 117°, we do the following:
117° + 360° = 477°
117° - 360° = 243°
As you can see, when we added 117° to 360°, we got 477°.
To answer your question, YES.
You now know that 117° and 477° are coterminal angles.
Keep in mind that 477° is not the only coterminal angle to 117°.
