Question

How do you determine the quadrant in which the angle $ -{{3.4}^{\circ }} $ lies?

Answer 1

Hint:
We start solving the problem by recalling the fact that that the quadrants are represented based on the values of the positive angles and the angles are measured positive if they make angle an angle with x-axis anti-clockwise. We then recall the fact that the angles are represented negatively if they are measured clockwise with respect to the x-axis. We then find the way to convert the negative angles to positive angles by adding $ {{360}^{\circ }} $ to it. We then check the quadrant in which the given angle lies to get the required answer.

Complete step by step answer:
According to the problem, we are asked to determine the quadrant in which the angle $ -{{3.4}^{\circ }} $ lies.
We know that the quadrants are represented based on the values of the positive angles. We know that the angles are measured positive if they make angle an angle with x-axis anti-clockwise.
So, the quadrants are marked as per the range of angles are given as follows: $ {{Q}_{1}}:{{0}^{\circ }}\le \theta \le {{90}^{\circ }} $ , $ {{Q}_{2}}:{{90}^{\circ }}\le \theta \le {{180}^{\circ }} $ , $ {{Q}_{3}}:{{180}^{\circ }}\le \theta \le {{270}^{\circ }} $ and $ {{Q}_{4}}:{{270}^{\circ }}\le \theta \le {{360}^{\circ }} $ ---(1).
Now, the angles are represented negative if they are measured clockwise with respect to the x-axis.
So, the range for quadrants will just be opposite for the angles with negative values which means that the equivalent positive angle can be found by adding $ {{360}^{\circ }} $ to the given negative angle.
Now, let us find the equivalent angle for given $ -{{3.4}^{\circ }} $ .
So, the equivalent angle is $ {{360}^{\circ }}-{{3.4}^{\circ }}={{356.6}^{\circ }} $ .
We can see that the angle $ {{356.6}^{\circ }} $ lies in the interval $ {{Q}_{4}}:{{270}^{\circ }}\le \theta \le {{360}^{\circ }} $ ., which means that the given angle lies in fourth quadrant.
 $ \, therefore, $ We have found that the given angle $ -{{3.4}^{\circ }} $ lies in the fourth quadrant.

Note:
 We can also solve this problem by taking a quadrant system and then marking the angle taking clockwise by following the properties of the angle to get the required answer. We should know that there are only four quadrants present in the system which constitute an angle of $ {{360}^{\circ }} $. Whenever we get the angle less than $ {{0}^{\circ }} $ , we try to add the multiples of $ {{360}^{\circ }} $ to get the equivalent positive angle which gives the required answer.