Question

How do you evaluate ${{\sin }^{2}}\left( \dfrac{\pi }{6} \right)$ ?

Answer 1

The term Trigonometry means “triangle” and “measure”. Trigonometry deals with especially right-angled triangles. We can find any value of the length of the triangle when angles are given or else, we can find the angle when the length of the triangle is given.
To know how to find the angle and its corresponding length,
Firstly, we need to know the two right triangles for finding the trigonometric values.

Complete step by step solution:
These two triangles are mainly used to find the trigonometric angles and their values.
Each trigonometric function has different ratios to find the lengths for particular angles.
The definition of Sine is the ratio of opposite upon hypotenuse.
In a similar way,
The definition of Cosine is the ratio of adjacent hypotenuses.
And the definition of Tangent is the ratio of opposite upon adjacent.
So, in the question, it is given that the angle is $\dfrac{\pi }{6}$ which is ${{30}^{\circ }}$
We also know that the definition of Sine is the ratio of opposite upon hypotenuse.
If we consider the triangle where the angles are ${{30}^{\circ }},{{60}^{\circ }}$ ,
The length opposite to the angle ${{30}^{\circ }}$ is $1$ and the hypotenuse is given by the length $2$
The ratio which is, opposite upon hypotenuse is thus given by, $\dfrac{1}{2}$
Hence the value of $\sin \dfrac{\pi }{6}$ or $\sin {{30}^{\circ }}$ is $\dfrac{1}{2}$
Since we asked to find, ${{\sin }^{2}}\dfrac{\pi }{6}$
It is nothing but the square of the value, $\sin \dfrac{\pi }{6}$
We have evaluated the value of $\sin \dfrac{\pi }{6}$ as $\dfrac{1}{2}$.
Hence the square of it will be,
$\Rightarrow \dfrac{1}{2}\times \dfrac{1}{2}$
$\Rightarrow \dfrac{1}{4}$
Hence the value of ${{\sin }^{2}}\dfrac{\pi }{6}$ is $\dfrac{1}{4}$

Note: One of the easier ways to memorize the trigonometric definitions are, Soh Cah Toa method.
The first word Soh depicts,
S for Sine, O for the opposite, H for the hypotenuse.
The second word Cah depicts,
C for Cos, A for the adjacent, H for the hypotenuse.
The third word Toa depicts,
T for Sine, O for the opposite, A for the adjacent.