Question

Find the angle of inclination of a straight line whose slope is $\dfrac{1}{\sqrt{3}}$.
(a) ${{60}^{\circ }}$
(b) ${{90}^{\circ }}$
(c) ${{30}^{\circ }}$
(d) ${{45}^{\circ }}$

Answer 1

When two straight lines intersect each other, then the space between these lines where they come in contact with each other, this space is called the Angle of Inclination. The angle of inclination is often represented in degrees. In order to find the angle of the inclination of a straight line, we need to use the formula that is used to find the slope of a straight line. The formula to calculate the slope of the straight line is given by $m=\tan \theta$ where m is the slope of the straight line and θ is the angle of inclination. Here, in this formula, we will replace the value of the slope, i.e., m with the value mentioned in the question. On replacing the value of m, we will evaluate the value of θ using the formula $\theta ={{\tan }^{-1}}\left( m \right)$. Hence, this will give the required value of the angle of inclination.

Complete step-by-step solution:
Here, we have been given with the slope of a straight line, i.e., $\dfrac{1}{\sqrt{3}}$. We have to find the angle of inclination of this straight line.
Let us suppose that m is the slope of the straight line. Then, according to the question,
$m=\dfrac{1}{\sqrt{3}}...............(i)$
Similarly, let us suppose that θ is the angle of inclination of the straight line.
Now, we will use the formula used to calculate the slope of a straight line. The formula for the inclination of the straight line is given by –
$m=\tan \theta ...........(ii)$
Then, using the equation (i) and (ii) and replacing the value of m from equation (i) in equation (ii), we get,
$\Rightarrow \dfrac{1}{\sqrt{3}}=\tan \theta$
Now, replacing the value of $\dfrac{1}{\sqrt{3}}$with equivalent trigonometric value in terms of tan, we get,
$\Rightarrow \tan \left( {{30}^{\circ }} \right)=\tan \theta$
On equating both sides, we can cancel the ‘tan’ term, hence, we get,
$\therefore \theta ={{30}^{\circ }}$
Hence, the angle of the inclination of a straight line is ${{30}^{\circ }}$.
Thus, the correct option is the option (c).

Note: While solving these types of questions, students normally makes mistake while keeping the trigonometric values. Students need to learn the numerical value of trigonometric values. Besides, students have to remember the formula to calculate the slope of a straight line. This formula can be used to find the slope of a straight line, provided the angle of inclination of the straight line is given. Students should be careful while replacing values and solving the equation. An alternate way to solve this can be –
Here, we have been given with the slope of a straight line, i.e., $\dfrac{1}{\sqrt{3}}$. We have to find the angle of the inclination of this straight line.
Let us suppose that m is the slope of the straight line. Then, according to the question,
$m=\dfrac{1}{\sqrt{3}}...............(i)$
Similarly, let us suppose that θ is the angle of the inclination of the straight line.
Now, we will use the formula used to calculate the slope of a straight line. The formula for the inclination of the straight line is given by –
$m=\tan \theta ...........(ii)$
Then, using the equation (i) and (ii) and replacing the value of m from equation (i) in equation (ii), we get,
$\Rightarrow \dfrac{1}{\sqrt{3}}=\tan \theta$
$\Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{1}{\sqrt{3}} \right)$
$\therefore \theta ={{30}^{\circ }}$