Question

Evaluate the value of tan 105 degrees \[\left( {\tan \,{{105}^ \circ }} \right)\].?

Answer 1

Hint: Here in this question, we have to find the exact value of a given trigonometric function by using the tangent sum or difference identity. First rewrite the given angle in the form of addition or difference, then the standard trigonometric formula tangent sum i.e., \[\tan \left( {A + B} \right)\] or tangent difference i.e., \[\tan \left( {A - B} \right)\] identity defined as \[\dfrac{{\tan A + \tan B}}{{1 - \tan A \cdot \tan B}}\] and \[\dfrac{{\tan A - \tan B}}{{1 + \tan A \cdot \tan B}}\] using one of these we get required value.

Complete step-by-step answer:
Consider the given trigonometric function
\[ \Rightarrow \,\,\tan {105^ \circ }\]-------(1)
The given trigonometric function has an angle \[{105^{^ \circ }}\], which is not a specified standard angle so we can’t tell the exact value directly. To find the value we have to convert the given angle in the form of sum or difference of specified angles.
The angle \[{105^ \circ }\] can be written as \[{60^ \circ } + {45^ \circ }\], then
Equation (1) becomes
\[ \Rightarrow \,\tan \left( {{{60}^ \circ } + {{45}^ \circ }} \right)\] ------(2)
Equation (2) looks similar as a trigonometric function \[\tan \left( {A + B} \right)\], then by Apply the trigonometric tangent sum identity is \[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A \cdot \tan B}}\].
Here \[A = {60^ \circ }\] and \[B = {45^ \circ }\]
Substitute A and B in formula, then
\[ \Rightarrow \,\,\tan \left( {{{60}^ \circ } + {{45}^ \circ }} \right) = \dfrac{{\tan {{60}^ \circ } + \tan {{45}^ \circ }}}{{1 - \tan {{60}^ \circ } \cdot \tan {{45}^ \circ }}}\]----------(3)
By using specified tangent angle from the standard trigonometric ratios table i.e., \[\tan \,6{0^ \circ } = \sqrt 3 \], and \[\tan {45^0} = 1\].
On, Substituting the values of angles in equation (3), we have
\[ \Rightarrow \,\,\tan \left( {{{105}^ \circ }} \right) = \dfrac{{\sqrt 3 + 1}}{{1 - \sqrt 3 \cdot 1}}\]
On simplification, we get
\[ \Rightarrow \,\,\tan \left( {{{105}^ \circ }} \right) = \dfrac{{\sqrt 3 + 1}}{{1 - \sqrt 3 }}\]
Hence, the exact functional value of \[\tan \left( {{{105}^ \circ }} \right) = \dfrac{{\sqrt 3 + 1}}{{1 - \sqrt 3 }}\].

Note: When the question is based on trigonometric function, we must know about the value for the trigonometry function so we need the table of trigonometry ratios for standard angles and also know the formulas like trigonometric identity, half and double angle formula, addition and difference identity of trigonometric function and transformation formulas.