Question

How do you find the value the exact value for \[\tan {120^{\circ}}\] ?

Answer 1

Hint: To evaluate the value of \[\tan {120^{\circ}}\]
As we all know that is a type of trigonometric function. So as we all know that
Gives positive values in the 1st as well as 3rd quadrant. The cot function also follows the same property. So in this question, we will use one property to convert an obtuse angle into an acute angle because we know standard values for acute angles. The property is
\[\tan \left( {{{90}^o} + \theta } \right) = - \cot \theta \]
By using this property we will try to evaluate the expression.

Step by step solution:
Firstly, we will convert obtuse angle into acute angle by using the below property
\[\tan \left( {{{90}^o} + \theta } \right) = - \cot \theta \]
Applying this property in original equation we get
\[ \Rightarrow \tan \left( {{{90}^o} + {{30}^o}} \right) = - \cot {30^o}\]
As we know that the standard value of \[\cot {30^o}{\rm{ }}is{\rm{ }}\sqrt 3 \]
So, by putting the value in the above equation we get
\[\tan \left( {{{120}^o}} \right) = - \sqrt 3 \]

Hence, our required answer is \[ - \sqrt 3 \]

Note:
We can convert an obtuse angle into an acute angle by another property that is
\[\tan \left( {{{180}^o} - \theta } \right) = - \tan \theta \]. But we need to take care that we must convert angles into acute angles so that we can apply our given standard result of a trigonometric function. In a similar way we can apply this type of property on other trigonometric functions that are sin, cos, sec, cosec but while using these trigonometric functions we should first observe which function is positive in which quadrant.