Question

Find the value of the trigonometric function: $\cot \left( -\dfrac{15\pi }{4} \right)$.

Answer 1

Hint: In this question, we can use the concept that tangent and cotangent ratios of an angle θ and of the sum of a multiple of π and θ \[\left( n\times \pi +\theta \right)\] are equal. So, here we may convert $\cot \left( -\dfrac{15\pi }{4} \right)$ into the form of $\cot \left( n\times \pi +\theta \right)$ where n is any integer, and then equalize it to $\cot \theta $ and get our required answer.

Complete step-by-step answer:
In this given question, we are asked to find the value of the trigonometric function: $\cot \left( -\dfrac{15\pi }{4} \right)$.

As we know, tangent and cotangent ratios of an angle θ and of the sum of a multiple of π and θ \[\left( n\times \pi +\theta \right)\] are equal. Statement……. (1.1).

So, here we can convert the given trigonometric ratio of $\cot \left( -\dfrac{15\pi }{4} \right)$ into the form of $\cot \left( n\times \pi +\theta \right)$ where n is any integer, that is $\cot \left( -\dfrac{16\pi }{4}+\dfrac{\pi }{4} \right)$.

It gives us $\cot \left( -\dfrac{15\pi }{4} \right)=\cot \left( -\dfrac{16\pi }{4}+\dfrac{\pi }{4} \right)..........(1.1)$

Now, as per statement 1.1 we can write equation 1.1 as:

$\cot \left( -\dfrac{15\pi }{4} \right)=\cot \left( -\dfrac{16\pi }{4}+\dfrac{\pi }{4} \right)=\cot \left( -4\pi +\dfrac{\pi }{4} \right)=\cot \left( \dfrac{\pi }{4} \right)..........(1.2)$, where we take n as -4.

Now, we know that the value of $\left( \dfrac{\pi }{4} \right)$ corresponds to ${{45}^{\circ }}$.

So, $\cot \left( \dfrac{\pi }{4} \right)=\cot {{45}^{\circ }}=1..........(1.3)$

Hence, from equation 1.4, we get the value of $\cot \left( \dfrac{\pi }{4} \right)=1$.

So, from equation 1.2, 1.3 and 1.4, we get $\cot \left( -\dfrac{15\pi }{4} \right)=1$.

Therefore, from equation 1.4 we have got our answer to the question as the value of $\cot
\left( -\dfrac{15\pi }{4} \right)$ as $1$.


Note: In this question, we could also have converted $\cot \left( -\dfrac{15\pi }{4} \right)$ into the form of $\left( -3\pi -\dfrac{3\pi }{4} \right)$. Here, the interesting thing is that this conversion would also have led us to the same answer as we have got.