Question

If $\tan \theta =\sqrt{2}$ , then the value of $\theta$ is:
a). less than $\dfrac{\pi }{4}$
b). equal to $\dfrac{\pi }{4}$
c). between $\dfrac{\pi }{4}$ and $\dfrac{\pi }{3}$
d). greater than $\dfrac{\pi }{3}$

Answer 1

To solve the question like above we will iterate through each of the above options and then check every option and the option which satisfies our question is the correct answer. We will also use the value of $\tan \dfrac{\pi }{4}=1$ , $\tan \dfrac{\pi }{3}=\sqrt{3}$ .

Complete step-by-step answer:
Since, we have to find the value of $\theta$ for which $\tan \theta =\sqrt{2}$ . To solve the question, we will iterate through each option.
Also, $\tan \theta =\sqrt{2}=1.414$
So, Option (a): It states that $\theta$ is less than $\dfrac{\pi }{4}$ , and we know that $\tan \dfrac{\pi }{4}=1$ , so we can say that if $\theta$ is less than $\dfrac{\pi }{4}$ , then $\tan \theta$ must be less than 1 because $\tan \theta$ is an increasing function between $\left( 0,\dfrac{\pi }{2} \right)$ .
But, it is given in the question that $\tan \theta =\sqrt{2}=1.414$ , which is greater than 1 so $\theta$ can’t be less than $\dfrac{\pi }{4}$ .
Hence, option (a) is incorrect.
Option (b): It states that $\theta$ is equal to $\dfrac{\pi }{4}$ . And, we have seen above in option (a) that $\tan \dfrac{\pi }{4}=1$ , but we have to find $\theta$ for which $\tan \theta =\sqrt{2}=1.414$ . So, option (b) is also incorrect.
Option (c): It states that $\theta$ is between $\dfrac{\pi }{4}$ and $\dfrac{\pi }{3}$ . And, we know that $\tan \dfrac{\pi }{4}=1$ and $\tan \dfrac{\pi }{3}=\sqrt{3}=1.732$ so, we can say that $\tan \theta$ must lies in between 1 and 1.732.
And, we have to find the value of $\theta$ for which $\tan \theta =\sqrt{2}=1.414$ and since 1.414 lies between 1 and 1.732. So, we can say that $\tan \theta$ must lies between $\dfrac{\pi }{4}$ and $\dfrac{\pi }{3}$ if $\tan \theta =\sqrt{2}$ .
Hence, option (c) is our correct option.
Option (d): It states that $\theta$ greater than $\dfrac{\pi }{3}$ . And, we know that $\tan \dfrac{\pi }{3}=\sqrt{3}=1.732$ , so we can say that $\tan \theta$ must be greater than 1.732 but we have to find such $\theta$ for which $\tan \theta =\sqrt{2}=1.414$ , and we can see that 1.414 is less than 1.732. So, $\theta$ can’t be greater than $\dfrac{\pi }{3}$ . So, option (d) is incorrect.
Hence, the correct option is only (c). This is our required answer.

So, the correct answer is “Option (c)”.

Note: Students are required to note that we directly do not know the value of $\theta$ for which $\tan \theta =\sqrt{2}$ from the normal trigonometric table which we generally remember. So, we can only calculate the range of the values, not the exact value.