Question

What is $\tan 180$ degrees?

Answer 1

Hint: To obtain the value of the trigonometric function given we will use its relation with other trigonometric functions. Firstly we know tangent is equal to sine divided by cosine so we will use this relation. Then as we know the value of $\sin {{180}^{\circ }}$ and $\cos {{180}^{\circ }}$ we will put it in the relation and get the value of $\tan {{180}^{\circ }}$ which is our desired answer.

Complete step by step answer:
We have to find the value of the given function:
$\tan 180$ Degrees
As we know that we can write tangent in form of sine and cosine as follows:
$\tan \theta =\dfrac{sin\theta }{\cos \theta }$……..$\left( 1 \right)$
We have our $\theta ={{180}^{\circ }}$
On putting the value of $\theta $ in equation (1) we get,
$\tan ~{{180}^{\circ }}=\dfrac{\sin {{180}^{\circ }}}{\cos {{180}^{\circ }}}$……$\left( 2 \right)$
As we know the value of sine and cosine are as follows:
$\begin{align}
  & \sin {{180}^{\circ }}=0 \\
 & \cos {{180}^{\circ }}=-1 \\
\end{align}$
Put above value in equation (2) and simply as follows,
$\begin{align}
  & \tan {{180}^{\circ }}=\dfrac{0}{-1} \\
 & \therefore \tan {{180}^{\circ }}=0 \\
\end{align}$

Hence $\tan 180$ degrees is 0.

Note: Trigonometric functions are obtained by the mean of a right-angled triangle sides and angles. There are six trigonometric functions which are sine, cosine, tangent, cosecant, secant and cotangent. All of them are related to each other in some way and there are various identities which involve these six trigonometric functions which are used to solve many mathematical problems. The three basic among them are sine, cosine and tangent as we can derive the last three by using these basic functions. As cosecant is reciprocal of sine, secant is reciprocal of cosine and cotangent is reciprocal of tangent. Sine, cosine, secant and cosecant have periodicity of $2\pi $ while tangent and cotangent have periodicity of $\pi $.