Question

Find the value of $\tan \left( {{{225}^ \circ }} \right)$.

Answer 1

We need to find the value of $\tan \left( {{{225}^ \circ }} \right)$. We see that we can write ${225^ \circ }$ as ${225^ \circ } = {180^ \circ } + {45^ \circ }$. Then, we know $\pi = {180^ \circ }$. After that we know, $\pi + \theta$ lies in third quadrant if $\theta < {90^ \circ }$ and $\tan \theta$ is positive if $\theta$ lies in the third quadrant. Also, $\tan \left( {\pi + \theta } \right) = \tan \theta$. So, we will find the value of $\tan \left( {{{225}^ \circ }} \right)$ using the above properties.

Complete step by step answer:
We need to find the value of $\tan \left( {{{225}^ \circ }} \right)$. Writing ${225^ \circ }$ as a sum of ${180^ \circ }$ and ${45^ \circ }$, we have
${225^ \circ } = {180^ \circ } + {45^ \circ }$
So, $\tan \left( {{{225}^ \circ }} \right) = \tan \left( {{{180}^ \circ } + {{45}^ \circ }} \right)$
As we know, $\pi = {180^ \circ }$, we can write
$\tan \left( {{{225}^ \circ }} \right) = \tan \left( {{{180}^ \circ } + {{45}^ \circ }} \right) = \tan \left( {\pi + {{45}^ \circ }} \right)$
As ${45^ \circ } < {90^ \circ }$, $\pi + {45^ \circ }$ lies in third quadrant and so $\tan \left( {\pi + {{45}^ \circ }} \right)$ will have positive value.
And, $\tan \left( {\pi + \theta } \right) = \tan \theta$. So, the equation becomes
$\Rightarrow \tan \left( {{{225}^ \circ }} \right) = \tan \left( {{{180}^ \circ } + {{45}^ \circ }} \right) = \tan \left( {\pi + {{45}^ \circ }} \right)$
Taking $\theta = {45^ \circ }$, we get
$\Rightarrow \tan \left( {{{225}^ \circ }} \right) = \tan \left( {{{180}^ \circ } + {{45}^ \circ }} \right) = \tan \left( {\pi + {{45}^ \circ }} \right) = \tan {45^ \circ }$
As we know the value of $\tan {45^ \circ }$, we will substitute the value.
Putting $\tan {45^ \circ } = 1$, we get
$\tan \left( {\pi + {{45}^ \circ }} \right)$
$= \tan {45^ \circ } = 1$

Therefore, we get $\tan \left( {{{225}^ \circ }} \right) = 1$.

Note: While we are finding the trigonometric value of a particular angle, we need to decompose in such a way that we know the value of the angle we will consider as $\theta$ in $\pi + \theta ,\dfrac{\pi }{2} + \theta ,\dfrac{{3\pi }}{2} + \theta$ and $2\pi + \theta$. Also, we need to consider that $\tan \left( {\pi + \theta } \right) = \tan \theta$, if $\pi + \theta$ lies in third or first quadrant and $\tan \left( {\pi + \theta } \right) = - \tan \theta$, if $\pi + \theta$ lies in second or fourth quadrant.