Question

How do you find the trig ratio $\sin \left( {{{30}^ \circ }} \right)$ ?

Answer 1

Hint : Here the question is related to the trigonometry, we use the trigonometry ratios and we are going to solve this question. By using the trigonometry properties we are going to solve this problem. To find the value we need the table of trigonometry ratios for standard angles.

Complete step-by-step answer :
The question is related to trigonometry and it includes the trigonometry ratios. The trigonometry ratios are sine, cosine, tan, cosec, sec, and cot.
We can solve this question by 2 methods
Method1:
Now consider the given question
$\sin \left( {{{30}^ \circ }} \right)$
This will lie in the first quadrant. In trigonometry we have ASTC rules for the trigonometry ratios. The above inequality lies in the first quadrant and the sine trigonometry ratio is positive in the first quadrant.
Now we consider the table
So we have a table for the trigonometry ratio sine for the standard angles.

Angle	0	30	45	60	90
sine	0	$\dfrac{1}{2}$	$\dfrac{1}{{\sqrt 2 }}$	$\dfrac{{\sqrt 3 }}{2}$	1

The above table is given for the standard degree. The given question is in the form of degree
therefore the value of $\sin ({30^ \circ })$ is $\dfrac{1}{2}$
Hence we have solved the trigonometry ratio and found the value.
Method 2:
Now consider an equilateral triangle. Each angle of an equilateral triangle measures ${60^ \circ }$ and since it is an equilateral triangle each side has length as 2a.
we had drawn a line perpendicular to the line BD from the point A. Now consider the triangle ABD.
The sine trigonometry ratio for the triangle ABD and for the angle A is defined and it is given by
$\sin \left( {{{30}^ \circ }} \right) = \dfrac{{opposite}}{{hypotenuse}}$
We know the value of the opposite side of the angle A and the hypotenuse of the angle A.
$\Rightarrow \sin \left( {{{30}^ \circ }} \right) = \dfrac{{BD}}{{AB}}$
The value BD is a because D is midpoint and the both sides are equal.
$\Rightarrow \sin \left( {{{30}^ \circ }} \right) = \dfrac{{BD}}{{AB}}$
on substituting know values we get
$\Rightarrow \sin \left( {{{30}^ \circ }} \right) = \dfrac{a}{{2a}}$
Cancelling the term a we get
$\Rightarrow \sin \left( {{{30}^ \circ }} \right) = \dfrac{1}{2}$
therefore the value of $\sin ({30^ \circ })$ is $\dfrac{1}{2}$
So, the correct answer is “ $\dfrac{1}{2}$ ”.

Note : In trigonometry to find the value of angles we have a table of trigonometry ratios for the standard angles. ASTC rule is applicable for the highest values. Whether the value of angle is in degree or radians the value for the standard angles will not change. Where ASTC rule is abbreviated as ALL SINE TANGENT COSINE.