The reference angle of 117° is
[tex]\theta=180^{\circ}-117^{\circ}=63^{\circ}.[/tex]
Therefore, we have to determine an angle in each quadrant with a reference angle of 63°. To illustrate the solution, we will use the following diagram as a reference:
From the above diagram, we get that:
[tex]\begin{gathered} \alpha=63^{\circ}, \\ \beta=180^{\circ}+63^{\circ}, \\ \gamma=360^{\circ}-63^{\circ}. \end{gathered}[/tex]
Simplifying the above results, we get:
[tex]\begin{gathered} \alpha=63^{\circ}, \\ \beta=243^{\circ}, \\ \gamma=297^{\circ}. \end{gathered}[/tex]
Finally, we get that, in the first quadrant the angle that has the same reference angle as 117° is:
[tex]63^{\circ},[/tex]
in the second quadrant is:
[tex]117^{\circ},[/tex]
in the third quadrant is:
[tex]243^{\circ},[/tex]
and in the fourth quadrant is:
[tex]297^{\circ.}[/tex]
Answer:
[tex]\begin{gathered} Quadrant\text{ I: 63}^{\circ}, \\ Quadrant\text{ II: 117}^{\circ}, \\ Quadrant\text{ III: 243}^{\circ}, \\ Quadrant\text{ IV: 297}^{\circ}. \end{gathered}[/tex]
