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}`