Question

How do you simplify $\tan ( - {35^\circ })$ using trigonometric identities?

Answer 1

Hint: Here to simplify the tangent of the angle, one must convert the tangent of the angle in a range between ${0^\circ }$ and ${180^\circ }$. For this, one can either convert $\tan $ to $\cot $ function or keep the tangent function as it is. Hence there is more than one way we can simplify $\tan {( - 35) ^\circ }$.

Complete step-by-step answer:
Here, we are asked to simplify the value of $\tan {( - 35) ^\circ }$ to simplify the calculations and plotting of the angle.
There are more than one ways to simplify the value, out of which we choose to continue using the tangent function.
We all know that the principal period of $\tan $ and $\cot $ functions is ${180^\circ }$ .
Also, the period of $\cot $ and $\tan $ functions can be said as ${360^\circ }$ , which is twice the value of the principal period ${180^\circ }$ .
Hence, adding ${360^\circ }$ won’t change the value of the angle.
$ \Rightarrow \tan {( - 35 + 360) ^\circ }$
$ \Rightarrow \tan {(325) ^\circ }$
Hence, we can say that the value of $\tan {(325) ^\circ }$ and $\tan ( - {35^\circ })$ is the same.
For an angle greater than ${270^\circ }$ , as cosine is positive and sine is negative, the value of $\cot $ will be negative in the $\;4th$ quadrant for which we can write the equation
$\Rightarrow$$\tan {(270 + \theta ) ^\circ } = - \cot \theta $ ,
Here, the value of $\theta $ can be obtained as
$ \Rightarrow 270 + \theta = 325$
$ \Rightarrow \theta = 325 - 270 = 55$
Hence, $\tan {(325)^\circ }$ can also be expressed as $ - \cot {55^\circ }$
As $\cot $ is an odd function,
$ \Rightarrow - \cot {55^\circ } = \cot {( - 55) ^\circ }$
Now, we know the principal period of the $\cot $ function is ${180^\circ }$ .
Hence adding an angle of ${180^\circ }$ won’t change the value of the function
$ \Rightarrow \cot {( - 55 + 180)^ \circ } = \cot {125^\circ }$
Hence, the angle $\tan ( - {35^\circ })$ can be written in a simplified way as $\tan {(325)^ \circ }$ , $\cot {( - 55)^ \circ }$ , and $\cot {125^\circ }$.

Note:
Here, we added twice the principal period to the given angle and got the above values. We can get a simplified angle $\tan {145^\circ }$ if we add the principal period to the given angle. Hence to represent an angle in a simplified way, there is no fixed value. All values can be obtained through each other.