Question

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

Answer 1

Hint: 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.