Question

How do I find the value of $ \text{cosec}{{225}^{\circ }} $ ?

Answer 1

Hint: In this question we need to find the value of $ \text{cosec}{{225}^{\circ }} $ . For this we will use the formula to find the value of reference angle and see where the angle $ {{225}^{\circ }} $ lie. We will see the sign of cosecant function in that quadrant and with the help of reference angle, we will find the value of $ \text{cosec}{{225}^{\circ }} $ . Value of the reference angle are known from the trigonometric ratio table.

Complete step by step answer:
Here we need to find the value of $ \text{cosec}{{225}^{\circ }} $ . From the trigonometric ratio table we do not know the value of $ \text{cosec}{{225}^{\circ }} $ , so we need to take help of a reference angle. We know that $ {{225}^{\circ }} $ can be written as \[{{180}^{\circ }}+{{45}^{\circ }}\] so we have $ \text{cosec}\left( {{225}^{\circ }} \right)=\text{cosec}\left( {{180}^{\circ }}+{{45}^{\circ }} \right) $ .
Therefore we have the reference angle as $ {{45}^{\circ }} $ which means we need to find the value of $ \text{cosec}{{45}^{\circ }} $ and determine the sign by looking at the quadrant. As we can see $ {{180}^{\circ }}+\theta $ will lie in the third quadrant. So let’s see if $ \text{cosec}\theta $ is positive or negative there. We know that,
In the first quadrant, all trigonometric functions are positive.
In the second quadrant, sine and cosecant functions are positive.
In the third quadrant, tangent and cotangent functions are positive.
In the fourth quadrant, cosine and secant function are positive.
So we see cosecant function is positive only in the first or second quadrant. Therefore, it is negative in the third quadrant. So we can say that $ \text{cosec}{{45}^{\circ }} $ will be negative.
Now let us find the value of $ \text{cosec}{{45}^{\circ }} $ from the trigonometric ratio table,
The trigonometric ratio table looks like this,

$ \theta \to $	$ {{0}^{\circ }} $	$ {{30}^{\circ }} $	$ {{45}^{\circ }} $	$ {{60}^{\circ }} $	$ {{90}^{\circ }} $
sine	0	$ \dfrac{1}{2} $	$ \dfrac{1}{\sqrt{2}} $	$ \dfrac{\sqrt{3}}{2} $	1
cosine	1	$ \dfrac{\sqrt{3}}{2} $	$ \dfrac{1}{\sqrt{2}} $	$ \dfrac{1}{2} $	0
tan	0	$ \dfrac{1}{\sqrt{3}} $	1	$ \sqrt{3} $	Not defined
sec	1	$ \dfrac{2}{\sqrt{3}} $	$ \sqrt{2} $	2	Not defined
cosec	Not defined	2	$ \sqrt{2} $	$ \dfrac{2}{\sqrt{3}} $	1
cot	Not defined	$ \sqrt{3} $	1	$ \dfrac{1}{\sqrt{3}} $	0

We can see that $ \text{cosec}{{45}^{\circ }} $ is equal to $ \sqrt{2} $ .
Therefore, $ \text{cosec}\left( {{225}^{\circ }} \right)=\text{cosec}\left( {{180}^{\circ }}+{{45}^{\circ }} \right)=-\text{cosec}{{45}^{\circ }}=-\sqrt{2} $ .
Hence the required value of $ \text{cosec}{{225}^{\circ }} $ is $ -\sqrt{2} $ .

Note:
 Students should take care of the signs while writing the final answer. Keep in mind the trigonometric ratio table for easily calculating the values. Also, keep in mind the signs of function in different quadrants. Use the acronym ASTC where A means all positive in the first quadrant, S means sine and its reciprocal (cosec) positive in second quadrant, I means tan and its reciprocal (cot) positive in third quadrant and C means cos and its reciprocal (sec) positive in the fourth quadrant.