Question

Find the value of the expression $\tan \left( -{{585}^{\circ }} \right)$

Answer 1

Hint: Use the formula $\tan \left( -\theta \right)=-\tan \theta $ to get the given expression in simpler form.

Complete step-by-step answer:
Convert the given angle in the expression of the problem to acute angle form i.e. between ${{0}^{\circ }}$ to ${{90}^{\circ }}\left( or\dfrac{\pi }{2} \right)$ . And use the quadrant rules that tan function is positive in the first and third quadrant and negative in the second and fourth quadrant. If the angle in tan function has involvement of multiple of $\dfrac{\pi }{2}\left( not\text{ }\pi \right)$ i.e. of type $\dfrac{n\pi }{2}\pm \theta $ then change tan function to cot or if summation of angle is of type $n\pi \pm \theta $ , then do not change the trigonometric function. Use the above rule to get the answer. Use $\tan \left( \dfrac{\pi }{4} \right)=1$

Given expression in the problem is

$\tan \left( -{{585}^{\circ }} \right)$

Let us suppose the value of given expression is ‘A’ so, we can write equation as

$A=\tan \left( -{{585}^{\circ }} \right)..........\left( i \right)$

Now, as the angle inside the expression is negative. So, we need to use the following trigonometric identity of tan functions, given as

$\tan \left( -\theta \right)=-\tan \theta $

So, using the above expression, we can rewrite the equation (i) as

$\begin{align}

  & A=\tan \left( -{{585}^{\circ }} \right)=-\tan {{585}^{\circ }} \\

 & A=-\tan {{585}^{\circ }} \\

\end{align}$

Now, we can observe that the angle involved in the above expression is not lying in ${{0}^{\circ }}$ to ${{90}^{\circ }}$ i.e. not acute angle and we have known values of trigonometric functions only in ${{0}^{\circ }}$ to ${{90}^{\circ }}$ . It means we have to convert the given angle to acute angle form with the help of some trigonometric identities.

Hence, let us divide the given expression by ${{180}^{\circ }}$ , so that we can write the given angle in form of sum of angle which is multiple of ${{180}^{\circ }}$ in following way:

$585=180\times 3+{{45}^{\circ }}$

Now, as we know that radian representation of ${{180}^{\circ }}$ is given as

$\pi $ radian = ${{180}^{\circ }}............\left( iii \right)$

And with the help of above relation, we can rewrite the angle ${{45}^{\circ }}$ in radian as well, given as

${{45}^{\circ }}=\dfrac{\pi }{4}radian..............\left( iv \right)$

Hence, we can rewrite the angle ${{585}^{\circ }}$ with the help of equation(iii) and (iv) as
$585=3\pi +\dfrac{\pi }{4}$

Now, we can rewrite the given expression in the problem i.e. equation(ii) as

$A=-\tan \left( 3\pi +\dfrac{\pi }{4} \right).........\left( vii \right)$

Now, as we know the quadrant angles are defined as

Now, we can apply the trigonometric rules for conversion of trigonometric expressions by changing its angle.
Now, we know the angle $\left( 3\pi +\dfrac{\pi }{4} \right)$ will lie in the third quadrant and as per the rules of trigonometric function in third quadrants, tan function is positive in 3rd quadrant. And as the difference of angle written in terms of multiple of $\pi $ $\left( 3\pi \right)$ , it means the trigonometric function will remain the same. And hence, the trigonometric relation for $\tan \left( 3\pi +\theta \right)$ can be given as

$\tan \left( 3\pi +\theta \right)=\tan \theta .........\left( viii \right)$

Hence, we get the equation(vii) from the result (viii) as

\[\begin{align}

  & A=-\tan \left( 3\pi +\dfrac{\pi }{4} \right)=-\tan \dfrac{\pi }{4} \\

 & A=-\tan \dfrac{\pi }{4} \\

 & A=-\tan \dfrac{\pi }{4}=-1 \\

\end{align}\]

Hence, the answer is -1.

Now, we know the value of $\tan \dfrac{\pi }{4}$ or $\tan {{45}^{\circ }}$ is given as 1. So, we get the value of A as

\[A=-\tan \dfrac{\pi }{4}=-1\]

Hence, the answer is -1.

Note: We need to know two important rules involved for conversion of trigonometric function with respect to the angles.
(i) Take care of the sign with the help of a given trigonometric function and the quadrant in which the angle is lying. This rule can be given as
(ii) If the angle involved inside trigonometric function is multiply of $\dfrac{\pi }{2}$ (not multiple of $\pi $ ) i.e. $\dfrac{n\pi }{2}\pm \theta $ type, where n is an odd integer, then we need to convert the
$\begin{align}
  & \sin \rightleftarrows \cos \\
 & \tan \rightleftarrows \cot \\
 & \sec \rightleftarrows \cos ec \\
\end{align}$
and if the angle involved in the sum is multiple of $\pi $ i.e. $n\pi \pm \theta $ type, then the trigonometric function will remain the same. Use the above two rules for conversion of any trigonometric function by changing their angles.
One may confuse the identity $\tan \left( -\theta \right)=-\tan \theta $ by other relations for other trigonometric functions. For future reference, other trigonometric relations are given as
$\begin{align}
  & \sin \left( -x \right)=-\sin x,\tan \left( -x \right)=-\tan x \\
 & \cos ec\left( -x \right)=-\cos ecx,\sec \left( -x \right)=\sec x \\
 & \cot \left( -x \right)=-\cot x \\
\end{align}$