Question

Find anangle in each quadrant with a common reference angle with 117°, from0°30<360°

Answer 1

The reference angle of 117° is

$\theta=180^{\circ}-117^{\circ}=63^{\circ}.$

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:

$\begin{gathered} \alpha=63^{\circ}, \\ \beta=180^{\circ}+63^{\circ}, \\ \gamma=360^{\circ}-63^{\circ}. \end{gathered}$

Simplifying the above results, we get:

$\begin{gathered} \alpha=63^{\circ}, \\ \beta=243^{\circ}, \\ \gamma=297^{\circ}. \end{gathered}$

Finally, we get that, in the first quadrant the angle that has the same reference angle as 117° is:

$63^{\circ},$

in the second quadrant is:

$117^{\circ},$

in the third quadrant is:

$243^{\circ},$

and in the fourth quadrant is:

$297^{\circ.}$

Answer:

$\begin{gathered} Quadrant\text{ I: 63}^{\circ}, \\ Quadrant\text{ II: 117}^{\circ}, \\ Quadrant\text{ III: 243}^{\circ}, \\ Quadrant\text{ IV: 297}^{\circ}. \end{gathered}$