Question

How do you find the exact value of \[\tan \left( {\dfrac{{3\pi }}{4}} \right)\]?

Answer 1

Hint:Here, we will rewrite the given angle as a difference of two angles and use the quadrants to determine that the given angle is exactly in which quadrant. Then we will apply the negative or positive sign according to the quadrant the tangent function belongs to. We will then use the trigonometric table to find the required value.

Complete step-by-step answer:
In order to find the value of \[\tan \left( {\dfrac{{3\pi }}{4}} \right)\]
Rewriting the given trigonometric angle, we get
\[\tan \left( {\dfrac{{3\pi }}{4}} \right) = \tan \left( {\pi - \dfrac{\pi }{4}} \right)\]
Now, we know that when we subtract any angle from \[\pi \] radians or \[180^\circ \], then we reach the second quadrant.
In the second quadrant, \[\tan \theta \] is negative.
Therefore, we can write \[\tan \left( {\dfrac{{3\pi }}{4}} \right)\] as:
\[\tan \left( {\dfrac{{3\pi }}{4}} \right) = - \tan \left( {\dfrac{\pi }{4}} \right)\]
Using the trigonometric table, we know \[\tan \left( {\dfrac{\pi }{4}} \right) = \tan 45^\circ = 1\].
\[ \Rightarrow \tan \left( {\dfrac{{3\pi }}{4}} \right) = - 1\]

Therefore, the exact value of \[\tan \left( {\dfrac{{3\pi }}{4}} \right)\] is \[ - 1\].

Hence, this is the required answer.

Additional information:
Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine, and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’, and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.

Note: An alternate way of solving this question is:
As we know, \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
Hence, in this question,
\[\tan \left( {\dfrac{{3\pi }}{4}} \right) = \dfrac{{\sin \left( {\dfrac{{3\pi }}{4}} \right)}}{{\cos \left( {\dfrac{{3\pi }}{4}} \right)}}\]
Now, using the unit circle, we know that, \[\sin \left( {\dfrac{{3\pi }}{4}} \right) = \dfrac{{\sqrt 2 }}{2}\] and \[\cos \left( {\dfrac{{3\pi }}{4}} \right) = - \dfrac{{\sqrt 2 }}{2}\].
Hence, substituting these values, we get,
\[ \Rightarrow \tan \left( {\dfrac{{3\pi }}{4}} \right) = \dfrac{{\dfrac{{\sqrt 2 }}{2}}}{{ - \dfrac{{\sqrt 2 }}{2}}}\]
Simplifying the expression, we get
\[ \Rightarrow \tan \left( {\dfrac{{3\pi }}{4}} \right) = \dfrac{{\sqrt 2 }}{2} \times - \dfrac{2}{{\sqrt 2 }} = - 1\]
Therefore, the exact value of \[\tan \left( {\dfrac{{3\pi }}{4}} \right)\] is \[ - 1\]
Hence, this is the required answer.