Question

The angle of measure \[1105^\circ \] lies in the
(a) first quadrant
(b) second quadrant
(c) third quadrant
(d) fourth quadrant

Answer 1

Hint:
Here, we need to find the quadrant where \[1105^\circ \] lies. We will use the fact that when increasing angles counterclockwise, the first quadrant comes again after the fourth quadrant. Since each quadrant measures \[90^\circ \], all angles between \[360^\circ \] and \[360^\circ + 90^\circ \] will lie in the first quadrant. We will keep adding \[90^\circ \] until we find the quadrant where the angle of measure \[1105^\circ \] lies.

Complete step by step solution:
We know that the first quadrant contains angles measuring between \[0^\circ \] and \[90^\circ \], the second quadrant contains angles measuring between \[90^\circ \] and \[180^\circ \], the third quadrant contains angles measuring between \[180^\circ \] and \[270^\circ \], and the fourth quadrant contains angles measuring between \[270^\circ \] and \[360^\circ \].
Now, we know that when increasing angles counterclockwise, the first quadrant comes again after the fourth quadrant.
For example: The angle measuring \[361^\circ \] also lies in the fourth quadrant.
Since each quadrant measures \[90^\circ \], all angles between \[360^\circ \] and \[360^\circ + 90^\circ \] will lie in the first quadrant.
Therefore, we get that the angles measuring between \[360^\circ \] and \[450^\circ \] also lie in the first quadrant.
Similarly, we can get the quadrants where the angels beyond \[450^\circ \] will lie.
The angles measuring between \[450^\circ \] and \[450^\circ + 90^\circ = 540^\circ \] will lie in the second quadrant.
The angles measuring between \[540^\circ \] and \[540^\circ + 90^\circ = 630^\circ \] will lie in the third quadrant.
The angles measuring between \[630^\circ \] and \[630^\circ + 90^\circ = 720^\circ \] will lie in the fourth quadrant.
Now, we know that the first quadrant will come again for the next angle measures.
The angles measuring between \[720^\circ \] and \[720^\circ + 90^\circ = 810^\circ \] will lie in the first quadrant.
The angles measuring between \[810^\circ \] and \[810^\circ + 90^\circ = 900^\circ \] will lie in the second quadrant.
The angles measuring between \[900^\circ \] and \[900^\circ + 90^\circ = 990^\circ \] will lie in the third quadrant.
The angles measuring between \[990^\circ \] and \[990^\circ + 90^\circ = 1080^\circ \] will lie in the fourth quadrant.
Finally, we do this once more to get the quadrant in which angle of measure \[1105^\circ \] lies.
The angles measuring between \[1080^\circ \] and \[1080^\circ + 90^\circ = 1170^\circ \] will lie in the first quadrant.
Therefore, we can observe that the angle of measure \[1105^\circ \] lies in the first quadrant.

Hence, the correct option is option (a).

Note:
We can also use the property that if an angle \[\theta \] lies in the \[{i^{{\rm{th}}}}\] quadrant, then the angle of measure \[\theta + 360^\circ n\] also lies in the \[{i^{{\rm{th}}}}\] quadrant. Here, \[i = 1,2,3,4\] and \[n\] is an integer.
This is a generalization and an easier formula to use.
The given angle is of measure \[1105^\circ \].
We will rewrite this angle as the sum of an angle between \[0^\circ \] and \[360^\circ \], and a multiple of \[360^\circ \].
Rewriting the angle measure, we get
\[\begin{array}{l}1105^\circ = 25^\circ + 1080^\circ \\ \Rightarrow 1105^\circ = 25^\circ + 3 \times 360^\circ \end{array}\]
Therefore, the angle of measure \[1105^\circ \] lies in the same quadrant (and same position) as the angle of measure \[25^\circ \].
Since the angle of measure \[25^\circ \] lies in the first quadrant, the angle of measure \[1105^\circ \] also lies in the first quadrant.