Question

How do you find the value of $\cos (\dfrac{{5\pi }}{6})$?

Answer 1

Hint: Cosine is a trigonometric function. It is the ratio of base and hypotenuse of the right-angled triangle. All trigonometric functions are periodic, that is, they give the same output if their period is added or subtracted in their input.

Complete step by step solution:
We know the value of $\cos \theta $ for different values of $\theta $ ($\theta $ is input for the function)
Using the formula $\cos (\dfrac{\pi }{2} + \theta ) = - \sin \theta $,
As we know the above formula,
We can easily split $\dfrac{{5\pi }}{6}$as $\dfrac{\pi }{2} + \dfrac{\pi }{3}$ for our operational use
Now, put $\dfrac{\pi }{2} + \dfrac{\pi }{3}$ in place of $\dfrac{\pi }{2} + \theta $ in the function $\cos (\dfrac{\pi }{2} + \theta )$,
We will get the result as $\cos (\dfrac{\pi }{2} + \dfrac{\pi }{3})$ , where $\theta = \dfrac{\pi }{3}$
Hence, now we can write
$\cos (\dfrac{\pi }{2} + \dfrac{\pi }{3})$=\[ - \sin (\dfrac{\pi }{3})\]= $ - \dfrac{{\sqrt 3 }}{2}$

Therefore, $\cos (\dfrac{{5\pi }}{6})$= $ - \dfrac{{\sqrt 3 }}{2}$

This is our answer for the given question

Note: Cosine is a trigonometric function. Its sign changes to negative as it crosses the mark of 90° and again changes to positive when it crosses the mark of 270°. That is why we put a negative sign in front of the sine when we add 90° to the input of the cosine. Apply the angle after you find the angle with equivalent trigonometric values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant. Adding any value inside the $\cos \theta $ function, the range will be from negative one to positive one. But if we add any number outside the $\cos \theta $ function, its range will change as well as its maxima and its minima will also change. But it is to be noted that the difference between the maxima and minima will remain constant, that is two.