Question

What does $\cos ec\left( {\dfrac{\pi }{2}} \right)$ equal ?

Answer 1

Hint : In the given problem, we are required to find the cosecant of a given angle using some simple and basic trigonometric compound angle formulae and trigonometric identities. Such questions require basic knowledge of compound angle formulae and their applications in this type of questions.

Complete step-by-step answer :
We can find out the value of $\cos ec\left( {\dfrac{\pi }{2}} \right)$ using the concepts of trigonometry and simplification using the arithmetic and algebraic rules.
Now, we know that $\dfrac{\pi }{2}$ radians correspond to $90$ degrees. We can also convert the angle from radians to degrees using the conversion $1$ radian equals to $180$ degrees. Radian is the standard unit for expressing the measurements of angles.
So, we have, $\cos ec\left( {\dfrac{\pi }{2}} \right)$ .
We know that the cosecant and sine trigonometric functions are reciprocal functions of each other. So, $\cos ecx = \dfrac{1}{{\sin x}}$ . So, we get,
$\Rightarrow \cos ec\left( {\dfrac{\pi }{2}} \right) = \dfrac{1}{{\sin \left( {\dfrac{\pi }{2}} \right)}}$
Now, we know that the value of the sine function for the angle $\left( {\dfrac{\pi }{2}} \right)$ radians is $1$ . Hence, we get the value of our expression as,
$\Rightarrow \cos ec\left( {\dfrac{\pi }{2}} \right) = \dfrac{1}{1} = 1$
Hence, we get the value of the trigonometric expression $\cos ec\left( {\dfrac{\pi }{2}} \right)$ as $1$ .
So, the correct answer is “1”.

Note : The problem given to us is a simple one involving the trigonometric functions. In case, the angle provided in the trigonometric functions is complex, then we would have to simplify the expression using the periodicity of the trigonometric functions. Periodic Function is a function that repeats its value after a certain interval. For a real number $T > 0$ , $f\left( {x + T} \right) = f\left( x \right)$ for all x. If T is the smallest positive real number such that $f\left( {x + T} \right) = f\left( x \right)$ for all x, then T is called the fundamental period. We must remember the value of trigonometric functions for some basic angles.