Question

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

Answer 1

In this question, we want to find the trigonometry angle value of the given equation between the intervals 0 to $2\pi$. Apply the formula ${\sin ^2}x = 2\sin x\cos x$.
Use the factorization method to solve the equation. At the end, we will find the value of the function. Based on those values, we will be able to find the value of the angle.

Complete step-by-step answer:
First, we will convert to degree using the conversational ratio$\dfrac{{180}}{\pi }$.
Here, we take only positive values to convert the ratio into degrees.
$\Rightarrow \dfrac{\pi }{6} \times \dfrac{{180}}{\pi }$
Let us simplify it.
$\Rightarrow 30$ Degrees
As we know, the value of $\sin 30 = \dfrac{1}{2}$.
If the degree contains a positive value, then we will go counter clockwise from the 0 degree axis.
However, if it contains a negative value, then we will go clockwise from the 0 degree axis.
So, we can state that 30 and -30 degrees measure the same thing. But they are situated in different directions and different quadrants.
Now, let us consider the negative sign back to the degree value. The ratio will be $- \dfrac{1}{2}$.
Hence, we must say that our ratio is in the fourth quadrant. And in the fourth quadrant the sine function value is always negative.
Therefore,

$\Rightarrow \sin \left( { - \dfrac{\pi }{6}} \right) = - \dfrac{1}{2}$

Note:
Here, we must remember the trigonometry ratios and the value of ratio at the angles 0, 30, 45, 60, and 90. And also know about trigonometry function in quadrant coordinate.
Some real-life application of trigonometry:

Used to measure the heights of buildings or mountains.
Used in calculus.
Used in physics.
Used in criminology.
Used in marine biology.
Used in cartography.
Used in a satellite system.