Question

Mark the position of the revolving line when it has traced out the following angles
A. $1185{}^\circ$
B. ${{150}^{g}}$

Answer 1

Hint: In order to solve this question, we have to convert the angles into degrees if they are not given in degrees. An angle in radians is converted into degrees by using the formula, $\text{Degree}=\text{Gradian}\times 0.9$. After that we will try to write the given angle in radian and then in the form of $n\pi +\theta$, and we will be able to trace the line.

Complete step-by-step answer:
In this question, we have been asked to trace the position of the revolving line of some angles. For that, we will first convert the angles into radians from degrees and then express it in the form of $n\pi +\theta$, where n is a natural number. So, let us consider each part of the question separately.
A. $1185{}^\circ$
Here, we will first convert $1185{}^\circ$ into radian, for that we will use the concept of $\pi \text{ }radians=180{}^\circ$. And we can also write it as,
$\begin{aligned} & \dfrac{\pi }{180{}^\circ }radians=1{}^\circ \\\ & \Rightarrow 1{}^\circ =\dfrac{\pi }{180{}^\circ }radians \\\ \end{aligned}$
Now, we will multiply both sides of the above equation by 1185, so we get,
$\begin{aligned} & 1185{}^\circ =\dfrac{\pi }{180}\times 1185\text{ }radians \\\ & \Rightarrow 1185{}^\circ =\dfrac{79\pi }{12}\text{ }radians \\\ \end{aligned}$
Now, we will express $\dfrac{79\pi }{12}$ in terms of $n\pi +\theta$. So, we will write the same as,
$\begin{aligned} & \dfrac{79\pi }{12}=\dfrac{72\pi +7\pi }{12} \\\ & \Rightarrow \dfrac{79\pi }{12}=6\pi +\dfrac{7\pi }{12}=3\times 2\pi +\dfrac{7\pi }{12} \\\ \end{aligned}$
So, now we can say that $1185{}^\circ =6\pi +\dfrac{7\pi }{12}\text{ }radians$. Now, for tracing the revolving line of the angle, we have to have a 2D graph, which has the left and right axis as negative and positive x axis, whereas the axis perpendicular to them is positive and negative y axis. As the angle is $6\pi +\dfrac{7\pi }{12}$ and we know that 1 revolution is completed at $2\pi$, it means that the line will complete 3 revolutions and then it will be either in the first quadrant or in the second quadrant, depending on the value of $\dfrac{7\pi }{12}$. We can write it as $\dfrac{7\pi }{12}=\dfrac{\pi }{2}+\dfrac{\pi }{12}$. Since it is in the form of $\dfrac{\pi }{2}+\theta$, it will be in the second quadrant at an angle of $\dfrac{\pi }{12}$ from positive y axis. Hence, we can represent it as line OA in the figure given below.

B. ${{150}^{g}}$
Here, we have been given an angle in the gradient and therefore to convert it to radians, we have to first convert it into degrees. We know that $\text{Degree}=\text{Gradian}\times 0.9$, so we can write it as,
$\begin{aligned} & {{150}^{g}}\text{ in degrees}=150\times 0.9 \\\ & \Rightarrow {{150}^{g}}\text{ in degrees}=135{}^\circ \\\ \end{aligned}$
Now, we will convert $135{}^\circ$ into radians to express it in $n\pi +\theta$ form. We know that $180{}^\circ =\pi \text{ }radians$, and therefore we can write it as, $1{}^\circ =\dfrac{\pi }{180}radians$. So, we can write,
$\begin{aligned} & 135{}^\circ =\dfrac{\pi }{180}\times 135\text{ }radians \\\ & \Rightarrow 135{}^\circ =\dfrac{3\pi }{4}\text{ }radians \\\ \end{aligned}$
Now, we will express $\dfrac{3\pi }{4}$ in terms of $n\pi +\theta$. So, we will write the same as,
$\begin{aligned} & \dfrac{3\pi }{4}=\dfrac{0\pi +3\pi }{4} \\\ & \Rightarrow \dfrac{3\pi }{4}=0+\dfrac{3\pi }{4} \\\ \end{aligned}$
Now, we can say that ${{150}^{g}}=\dfrac{3\pi }{4}radians$. Now, for tracing the line, we can see that it will either lie in the first quadrant or in the second quadrant depending on $\dfrac{3\pi }{4}$. We can write it as $\dfrac{3\pi }{4}=\dfrac{\pi }{2}+\dfrac{\pi }{4}$. Since it is in the form of $\dfrac{\pi }{2}+\theta$, it will be in the second quadrant, at an angle of $\dfrac{\pi }{4}$ from the positive y axis. Hence, we can represent the line as line OB as in the figure given below.

Note: Here, we have expressed angles in the form of $n\pi +\theta$, because in the coordinate axis we have four quadrants and after an angle of $360{}^\circ$, we again reach the first quadrant and then the angles start repeating. This makes it easier to understand and represent the angles in the correct quadrants. So, care must be taken not to make mistakes in this portion.