Question

What is $\tan (-{{585}^{\circ }})$ equal to?
A. 1
B. -1
C. $-\sqrt{2}$
D. $-\sqrt{3}$

Answer 1

Hint: We can use negative angle property of tan as given below:
$\tan (-\theta )=-\tan (\theta )$ also use the some conversion for angle in standard angles like $\tan ({{180}^{\circ }}+\theta )=\tan (\theta ) $ and $\tan \left( 2n\pi +\theta \right)=\tan (\theta ) $.

Complete step-by-step solution -
And to write ${{585}^{\circ }}$ in standard value which we know we need to use $\tan \left( 2n\pi +\theta \right)=\tan (\theta )$
 Given trigonometric ratio is $\tan (-{{585}^{\circ }})$
We can use first $\tan (-\theta )=-\tan (\theta )$
$\Rightarrow \tan (-{{585}^{\circ }})=-\tan ({{585}^{\circ }})$
Now we can write ${{585}^{\circ }}$ as ${{585}^{\circ }}={{360}^{\circ }}\times 1+{{225}^{\circ }}$
So we will get
$\Rightarrow \tan (-{{585}^{\circ }})=-\tan ({{360}^{\circ }}\times 1+{{225}^{\circ }})$
$\Rightarrow \tan (-{{585}^{\circ }})=-\tan (2\pi \times 1+{{225}^{\circ }})$ $\left\{ \because 2\pi ={{360}^{\circ }} \right\}$
$\Rightarrow \tan (-{{585}^{\circ }})=-\tan ({{225}^{\circ }})$ $\left\{ \because \tan \left( 2n\pi +\theta \right)=\tan (\theta ) \right\}$
$\Rightarrow \tan (-{{585}^{\circ }})=-\tan ({{180}^{\circ }}+{{45}^{\circ }})$ $\left\{ \because \tan ({{180}^{\circ }}+\theta )=\tan (\theta ) \right\}$
$\Rightarrow \tan (-{{585}^{\circ }})=-\tan ({{45}^{\circ }})$
$\Rightarrow \tan (-{{585}^{\circ }})=-1$
Hence option B is correct.

Note: In this question, we need to be careful about how to write an angle as a sum of two angles. We always write it in that way from which we can easily convert a given angle in standard angle values. Standard angle values are ${{0}^{\circ }},{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }},{{90}^{\circ }}$.