Given:
The value is tan(195°).
To find:
The exact value of tan(195°).
Solution:
We have,
[tex]\tan (195^\circ)=\tan (180^\circ+15^\circ)[/tex]
[tex]\tan (195^\circ)=\tan (15^\circ)[/tex] [tex][\because \tan (180^\circ-\theta)=\tan \theta][/tex]
It can be written as
[tex]\tan (195^\circ)=\tan (45^\circ-30^\circ)[/tex]
[tex]\tan (195^\circ)=\dfrac{\tan (45^\circ)-\tan (30^\circ)}{1+\tan (45^\circ)\tan (30^\circ)}[/tex] [tex][\because \tan (A-B)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}][/tex]
[tex]\tan (195^\circ)=\dfrac{1-\dfrac{1}{\sqrt{3}}}{1+(1)(\dfrac{1}{\sqrt{3}})}[/tex]
[tex]\tan (195^\circ)=\dfrac{\dfrac{\sqrt{3}-1}{\sqrt{3}}}{1+\dfrac{1}{\sqrt{3}}}[/tex]
[tex]\tan (195^\circ)=\dfrac{\dfrac{\sqrt{3}-1}{\sqrt{3}}}{\dfrac{\sqrt{3}+1}{\sqrt{3}}}[/tex]
[tex]\tan (195^\circ)=\dfrac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]
Therefore, the exact value of tan(195°) is [tex]\dfrac{\sqrt{3}-1}{\sqrt{3}+1}[/tex].
