Question

What is the value of $\tan \left( {{{180}^ \circ } - \theta } \right)$ ?

Answer 1

The given question deals with basic simplification of trigonometric function by using some of the simple trigonometric formulae. We must know the compound angle formula for tangent trigonometric function $\tan \left( {A - B} \right) = \dfrac{{\tan A - \tan B}}{{1 + \tan A\tan B}}$. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete step by step answer:
In the given problem, we have to simplify the expression $\tan \left( {{{180}^ \circ } - \theta } \right)$ and find its value.
So, we use the compound angle formula of tangent $\tan \left( {A - B} \right) = \dfrac{{\tan A - \tan B}}{{1 + \tan A\tan B}}$ in order to simplify the value of the expression given to us.
So, we have, $\tan \left( {{{180}^ \circ } - \theta } \right)$
Here, $A = {180^ \circ }$ and $B = \theta$.
$\Rightarrow \tan \left( {{{180}^ \circ } - \theta } \right) = \dfrac{{\tan {{180}^ \circ } - \tan \theta }}{{1 + \tan {{180}^ \circ }\tan \theta }}$
Now, we know that the value of $\tan {180^ \circ }$ is equal to zero. Hence, we get,
$\Rightarrow \tan \left( {{{180}^ \circ } - \theta } \right) = \dfrac{{0 - \tan \theta }}{{1 + \left( 0 \right)\tan \theta }}$
Now, simplifying the expression, we get,
$\therefore \tan \left( {{{180}^ \circ } - \theta } \right) = - \tan \theta$

So, the value of $\tan \left( {{{180}^ \circ } - \theta } \right)$ is equal to $- \tan \theta$.

Additional information:
Trigonometric functions are also called Circular functions. Trigonometric functions are the functions that relate an angle of a right angled triangle to the ratio of two side lengths. There are $6$trigonometric functions, namely: $\sin (x)$, $\cos (x)$, $\tan (x)$,$\cos ec(x)$, $\sec (x)$ and $\cot \left( x \right)$ . Also, $\cos ec(x)$ ,$\sec (x)$ and $\cot \left( x \right)$are the reciprocals of $\sin (x)$, $\cos (x)$ and $\tan (x)$ respectively.

Note: Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as: $\tan \left( {A - B} \right) = \dfrac{{\tan A - \tan B}}{{1 + \tan A\tan B}}$ and $\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$ . Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such type of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations.However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers.