Question

How do you find the exact value of $\tan 330^\circ$ ?

Answer 1

Hint: In this question, we have to find the value of the tangent of 330 degrees, tangent is a trigonometric function. So we must know the definition of trigonometric functions and how to find their values to solve the question. The relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse and one of the angles other than the right angle is studied using trigonometry. A pattern is repeated by the trigonometric functions after 360 degrees, as the given angle is smaller than 360 degrees but near to it, so we subtract 330 degrees from 360 degrees, and then find the tangent of the angle obtained.

Complete step by step answer:
We will find the solution,
$
\Rightarrow \tan (330^\circ ) = \tan [360^\circ + ( - 30^\circ ] \\
   \Rightarrow \tan (330^\circ ) = \tan ( - 30^\circ ) \\
 $
Now $\tan ( - 30^\circ )$ lies in the fourth quadrant, we know that tangent is negative in the fourth quadrant, so –
$
\Rightarrow \tan (330^\circ ) = - \tan (30^\circ ) \\
   \Rightarrow \tan (330^\circ ) = - \dfrac{1}{{\sqrt 3 }} \\
 $
Hence, the exact functional value of $\tan (330^\circ )$ is $ - \dfrac{1}{{\sqrt 3 }}$.

Note: In the first quadrant, the value of all the trigonometric functions is positive. The sine function is positive while all the other functions are negative In the second quadrant, the tangent function is positive while all other functions are negative in the third quadrant, and the cosine function is positive while all others are negative in the fourth quadrant, that’s why $\tan (360 - \theta )\,or\,\tan ( - \theta ) = - \tan \theta $. To solve the question related to trigonometry, we must know the value of trigonometric ratios of the basic angles like $0,\,\dfrac{\pi }{6},\,\dfrac{\pi }{4},\,\dfrac{\pi }{3},\,and\dfrac{\pi }{2}$.