Answer

Verified

389.4k+ views

**Hint:**We are given $\tan \left( \dfrac{3\pi }{4} \right)$ , we are asked to find its value, to do so we will learn about the relation between degree and the radian, then we will learn in which quadrant our coordinate will lie, once we have the quadrant will work on what sign it posses then using the value of the function in quadrant ‘I’, we will access the value of the function in that quadrant.

**Complete step by step answer:**

We are given $\tan \left( \dfrac{3\pi }{4} \right)$, we can see that the angle given to us is in radian.

We will change it into a degree as the degree is easy to see and understand.

Now the relation between degree and radian is given as –

180 degree $=\pi $ radian

Or

$\pi \text{ radian}=180\text{ degree}$

So, $1\text{ radian}=\dfrac{{{180}^{\circ }}}{\pi }\text{degree}$

By unitary method,

So, $\dfrac{3\pi }{4}\text{radian}=\dfrac{{{180}^{\circ }}}{\pi }\times \left( \dfrac{3\pi }{4} \right)\text{degree}$

By simplifying, we get –

$\dfrac{3\pi }{4}\text{radian}=135\text{ degree}$

Now we get the angle in degree is ${{135}^{\circ }}$ and we know that the first Quadrant is from 0 to ${{90}^{\circ }}$ .

second is from ${{90}^{\circ }}$ to ${{180}^{\circ }}$ so our angle ${{135}^{\circ }}$ will lie in the second Quadrant.

Now we will look for the sign of the tan functions in this Quadrant.

The tan function is positive in first and third Quadrant and in rest two it is negative so in Quadrant second, tan is negative.

Now we will see that ${{135}^{\circ }}$ can be written as ${{135}^{\circ }}={{180}^{\circ }}-{{45}^{\circ }}$ .

So, $\tan \left( {{135}^{\circ }} \right)=\tan \left( {{180}^{\circ }}-{{45}^{\circ }} \right)$

Now, we will use that –

$\tan \left( {{180}^{\circ }}-\theta \right)=-\tan \theta $

So for $\tan \left( {{180}^{\circ }}-{{45}^{\circ }} \right)=-\tan \left( {{45}^{\circ }} \right)$

As we know that $\tan {{45}^{\circ }}=1$

So,

$\begin{align}

& \tan \left( {{180}^{\circ }}-{{45}^{\circ }} \right)=-\tan {{45}^{\circ }} \\

& =-1 \\

\end{align}$

Hence, we get –

$\tan \left( {{135}^{\circ }} \right)=-1$

Or we can say that –

$\tan \left( \dfrac{3\pi }{4} \right)=-1$

**Note:**Remember it is easy to see this on the degree scale, but it is not necessary to always change to the degree scale, we can also solve on the radian form also.

Now $\dfrac{3\pi }{4}=\pi -\dfrac{\pi }{4}$

So, $\tan \left( \dfrac{3\pi }{4} \right)=\tan \left( \pi -\dfrac{\pi }{4} \right)$

As we know $\tan \left( \pi -\theta \right)=-\tan \theta $

So in our case $\theta =\dfrac{\pi }{4}$

So, $\tan \left( \pi -\dfrac{\pi }{4} \right)=-\tan \left( \dfrac{\pi }{4} \right)$

As $\tan \dfrac{\pi }{4}=1$

So,

$\tan \left( \dfrac{3\pi }{4} \right)=\tan \left( \pi -\dfrac{\pi }{4} \right)=-\tan \dfrac{\pi }{4}=-1$

Hence the value of $\tan \left( \dfrac{3\pi }{4} \right)$is -1.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE